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06 - Sequences & Series

Sequences & Series

Sequence

Quote

A Sequence is an arrangement of numbers subject to some definite rule.

OR

A Sequence is a function whose domain is a subset is the set if natural numbers.

  • Notation used for Sequence is \(\{a_n\}\).
  • Each number in the Sequence is called a term.
  • The nth term are or general term of a sequence is denoted by \(a_n\).
  • If all members of a sequence are real numbers, then it is called real sequence.
  • If domain of sequence is a finite set then the sequence is called a finite sequence otherwise infinite sequence.
  • An infinite sequence has no last term.
  • A series is obtained by adding or subtracting the terms of a sequence.
  • If the terms of a sequence follow certain pattern then the sequence is called a progression.

Types of Progression

  • Arithmetic Progression (AP)
  • Geometric Progression (GP)
  • Harmonic Progression (HP)

Arithmetic Progression

  • A sequence whose terms increase or decrease by a fixed number is called Arithmetic Progression.
  • The fixed number is called common difference of the A.P
  • Common difference (\(d\)) can be a +ve or -ve number.
  • Total term (\(n\)) can never be a negative number
  • General term of AP is: $$ a_n=a+(n-1)d $$

Properties of AP

  • If \(a_1\), \(a_2\), \(2_3\),..., \(a_n\) are in AP then:
    • \(a_1+k\), \(a_2+k\), \(a_3+k\),..., \(a_n+k\) are also in A.P
    • \(a_1-k\), \(a_2-k\), \(a_3-k\),..., \(a_n-k\) are also in A.P
    • \(a_1\cdotp k\), \(a_2\cdotp k\), \(a_3\cdotp k\),..., \(a_n\cdotp k\) are also in A.P
    • \(a_1/k\), \(a_2/k\), \(a_3/k\),..., \(a_n/k\) are also in A.P , \(k\rlap{/}{=}0\)
  • If \(a_1\), \(a_2\), \(2_3\),..., \(a_n\) are in A.P then:
    • In an A.P. of finitely many terms, sum of terms equidistant from the beginning and end is constant equal to the sum of the first and last terms and so on. $$ a_1 +a_n =a_2+ a_{n-1} =a_3+ a_{n-2} $$
    • \(a_n=(a_{r-k}+ a_{r+k}), 0\le k\le n-r\)
      For Example: \(2\), ? , ? , ? ,\(10\),\(12\) then \(a_3=?\)
      \(a_3=(a_{3-2} + a_{3+2})/2\)
      \(a_3= (a_1+a_5)/2\) \(a_3=(2+10)/2=6\)
  • If \(a^2\), \(b^2\), \(c^2\) are in AP then \(\frac{1}{b+c}\), \(\frac{1}{c+a}\), \(\frac{1}{a+b}\) are also in AP.
  • If \(l\),\(m\),\(n\) be the pth, qth and rth terms of an A.P then: $$ l(q-r)+m(r-p)+n(p-q)=0 $$ $$ p(m-n)+q(n-l)+r(l-m)=0 $$

Info

  • If we have to take three numbers in AP we take these to be $$ a-d, a, a+d $$
  • If we have to take four numbers in AP we take these to be $$ a-3d, a-d, a+d, a+3d $$
  • If we have to take five numbers in AP we take these to be $$ a-2d, a-d, a, a+d, a+2d $$

Arithmetic Mean

  • If three quantities are in AP then middle term is known as arithmetic mean (\(AM\)) of the other two.
  • If \(a\), \(A\), \(b\) are in AP then \(A=\frac{a+b}{2}\)
  • If \(a\), \(A_1\), \(A_2\), \(A_3\), ..., \(A_n\), \(b\) are in AP then the numbers \(A_1\), \(A_2\), \(A_3\), ..., \(A_n\) are called \(n\) arithmetic means between \(a\) and \(b\).
  • The sum of n AM between any two numbers \(a\) & \(b\) is equal to n-times the AM between \(a\) & \(b\) that is given by : $$ n\left(\frac{a+b}{2}\right) $$
  • nth arithmetic mean between \(a\) & \(b\) is given by $$ A_n=\frac{a+nb}{n+1} $$
  • mth AM from n terms in AP eg. finding \(A_3\) from \(2\), \(A_1\), \(A_2\), \(A_3\), \(A_4\), \(30\), we can use the formula $$ A_m=a+m\frac{b-a}{n+1} $$

Arithmetic Series

If \(a\), \(a+d\), \(a+2d\), \(a+3d\), ..., \(a+(n-1)d\) is an AP. Then series $$ a+(a+d)+(a+2d)+...+(a+(n-1)d) $$ is called an Arithmetic series.

Important Notes

  • If \(a_m\) and \(a_n\) are two terms of an \(AP\) whereas \(a_m\) is bigger then \(a_n\) also \(m>n\) then $$ d=\frac{a_m-a_n}{m-n} $$
  • If \(a\) is the first term and \(d\) be the common difference of an A.P having \(m\) terms then nth term from the end is \((m-n+1)\)th term from the beginning. Thus nth term from the end is given by $$ a_n=a_1+(m-n)d $$ whereas \(m\) is total terms and \(n\) is specific term from the end
  • If \(1/a\), \(1/b\), \(1/c\) are in A.P then \(b=2ac/a+c\).
  • If \(1/a\), \(1/b\), \(1/c\) are in A.P then \(d=a-c/2ac\)
  • For total no of terms in AP we use the formula $$ n=\frac{a_n-a_1}{d}+1 $$
  • Middle term of three consecutive terms of A.P. is A.M. between the extreme terms

Sum of Arithmetic Series

  • If any three of \(a\), \(n\), \(d\) or \(S_n\) is given use formula $$ S_n=\frac{n}{2} \left[2d+(n-1)d\right] $$
  • If first term (\(a_1\)) and last/general term (\(a_n\)) is given given then use formula $$ S_n=\frac{n}{2}[a_1+a_n] $$
  • Sum of Alternating Series:
    • \(a-a+\)\(a-a+a\)... having even number of terms is equal to \(0\)
    • \(a-a+\)\(a-a+a\)... having odd number of terms is equal to \(a\)

Shortcut for Sum

Once we get the corresponding terms for any A.P, we can easily find the sum of an A.P by using the property of averages. $$ S= n \times \text{A.M of that A.P} $$

For Example Sum of \(2\),\(6\),\(10\),\(14\),\(18\),\(22\)??
As, AM \(=(22+2)/2=12\)
So, Sum = \(6\times 12=72\)

Important Notes

  • If nth term of a sequence is a linear expression in n then sequence is an AP. eg. \(a_n=2n-1\) is an AP.
  • If the sum of the first n term of a sequence is a quadratic expression in n then the sequence is an A.P. eg. \(a_n=n(5n-1)\) is an quadratic expression.
  • When the middle term of A.P is given and no. of total terms are odd then sum can be given by $$ S_n=a_{middle}\times n $$
  • When the no. of total terms are even and the middle terms are two then Sum can be given by: $$ S_n=(\text{sum of middle terms}/2)\times n $$
  • When \(d=0\) then \(S_n=a+a+... n\) terms \(=na\)

Guessing Sum of AP

Here is a tip for guessing the sum of AP very quickly.

For example: \(1+\)\(3+\)\(5+\)\(7+\)...\(+(2n-1)=\)?

  • \(n^2\)
  • \(n(n+1)\)
  • \(2n+1\)
  • None of these

Simply put the some values of \(n\) and check the validity.

Put \(n=1\)

  • \(1^2\) \(=1\)
  • \(1(1+1)\) \(2\)
  • \(2(1)+1\) \(3\)

So, \(n^2\) is correct ans.

Geometric Progression

  • A Sequence of numbers in which every term after the first is obtained from the preceding term by multiplying it with a constant number is called a Geometric Progression
  • The general form of a GP is $$ a, ar, ar^2, ar^3, ..., ar^{n-1} $$
  • The general term of GP is $$ a_n=ar^{n-1} $$
  • The constant ratio of any two constant terms in GP is called by Common Ratio. It can be found by $$ \frac{a_n}{a_{n-1}} $$
  • Common Ratio can never be zero or one.
  • If a GP has n-terms then nth term is called last term or limiting term and it is denoted by \(l\) & is given by \(l=ar^{n-1}\)
  • If \(a_m\) and \(a_n\) are two terms of an GP whereas \(a_m>a_n\) also \(m>n\). Then $$ r=\left(\frac{a_m}{a_n}\right)^{\frac{1}{m-n}} $$
  • When last term \(l\) and common ratio \(r\) is given then: $$ a_n= \frac{l}{r^{n-1}} $$
  • If first term (\(a\)) and common ratio is given the nth term from last can be given by \(ar^{m-n}\)
  • Number of terms can be calculated by using formula $$ r^n = \frac{a_n}{a_1} r $$

Info

  • If we have to take three numbers in GP we take these to be $$ \frac{a}{r}, a, ar $$
  • If we have to take four numbers in GP we take these to be $$ \frac{a}{r^3}, \frac{a}{r}, ar, ar^3 $$
  • If we have to take five numbers in GP we take these to be $$ \frac{a}{r^2}, \frac{a}{r}, a, ar, ar^2 $$

Properties

  • After applying any basic mathematic operation on each of its term the progression remains in GP.
  • If we have two GP, the progression formed by the product or division of its corresponding terms, is also in GP.
  • In a G.P. of finitely many terms, the product of terms equidistant from the beginning and end is constant equal to the sum of the first and last terms.

Geometric Mean

  • If three quantities are in GP then middle term is known as geometric means of the other two.
  • If \(a\), \(G\), \(b\) are in GP, then \(G=\pm\sqrt{ab}\).
  • If \(a\), \(G_1\), \(G_2\),..., \(G_n\), \(b\) then the numbers \(G_1\), \(G_2\),..., \(G_n\) are called n geometric means between \(a\) and \(b\).
  • nth GM between \(a\) & \(b\) is given by $$ G_n=a\left(\frac{b}{a}\right)^{\frac{n}{n+1}} $$
  • The product if \(n\) geometric means between two given number is equal to nth power to single GM between these numbers. \(n\) GM between \(a\) & \(b\) is \(G_n=(\sqrt{ab})^n\) or \(a^n(b/a)^{n/2}\)
  • If \(A\) & \(G\) are respectively arithmetic and geometric means between two positive numbers \(a\) and \(b\) then:
    • \(A>G\)
    • The quadratic equation having \(a\) & \(b\) as its roots is \(x^2-2Ax+G^2=0\)
    • The two positive numbers are \(A\) & \(\pm\sqrt{A^2-G^2}\)

Geometric Series

If \(a\), \(ar\), \(ar^2\),..., \(ar^{n-1}\) is a GP then $$ a+ar+ar^2 +ar^3 +...+ ar^{n-1} $$ is called Geometric Series.

Sum of \(n\) terms of Geometric Series

Following are some formulas of Sum of Geometric Series: $$ S_n=\frac{a(1-r^n)}{1-r}, |r|<1 $$

\[ S_n=\frac{a-lr}{1-r}, |r|<1 \]
\[ S_n=\frac{a(r^n-1)}{r-1}, |r|>1 \]

$$ S_n=\frac{lr-a}{r-1}, |r|>1 $$ where \(l\) is last term of GP.

MCQ Trick

Finding total covered by the object dropped from height \(h\) and rebounds \(r\)th of its actual height. $$ \text{distance}=\frac{h(1+r)}{1-r} $$

Sum of infinite Geometric Series

The series \(a+\)\(ar^2+\)\(ar^3+\)...\(+ar^{n-1}+\)...\(\infty\) is called infinite geometric series. We can compute the sum only if \(|r|<1\)

Case 1 \(|r|<1\)

If for an infinite geometric series \(|r|<1\), the series is said to be Convergent Series. In this case $$ S_\infty=\frac{a_1}{1-r} $$

Sum of infinite Geometric Series when common ratio and first term is given can be found by: $$ S_\infty=\frac{1}{r}\times a_1 $$

Case 2 \(|r|>1\)

If for an infinite geometric series \(|r|>1\), the series is said to be Divergent Series.

In this case sum of series increases indefinitely so: $$ S_\infty \to \infty $$

Case 3 \(|r|=1\)

In this case also sum of series increases indefinitely, but we can calculated its sum if its is not infinite ie. \(n\) is given Thus $$ S_n=na_1 $$

Case 3 \(|r|=-1\)

If for an infinite geometric series \(r=-1\), then series is said to be Oscillatory Series. In this case $$ S_n=\frac{a_1-(-1)^na_1}{2} $$

Thus:

  • \(S_n=a_1\) if \(n\) is positive odd number
  • \(S_n=0\) if \(n\) is positive even number.

Arithmetic Geometric Progression (AGP)

  • It is the combination of Arithmetic & Geometric Progressions.
  • General form of AGP is $$ a, (a+d)r, (a+2d)r^2, (a+3d)r^3, ... $$
  • nth term of an AGP is $$ T_n=[a+(n-1)d]r^{n-1} $$

Note

A sequence is both an A.P. and a G.P. iff it is a constant sequence.

Harmonic Progression

A sequence of numbers is called a harmonic progression if the reciprocals of its terms are in arithmetic progression.

  • No term of HP can be zero
  • Every HP has a corresponding AP
  • There is no general method to find the sum of a HP.

Harmonic Mean

  • If three quantities are in HP then the middle is called harmonic mean of other two.
  • If \(a\), \(H\), \(b\) are in HP then $$ H=\frac{2ab}{a+b} $$
  • nth harmonic mean between \(a\) & \(b\) is given by $$ H_n=\frac{(n+1)ab}{na+b} $$

Relation between AP, GP & HP

  • \(G^2=AH\)
  • \(A>G>H\), if \(a\) & \(b\) are two distinct +ve real numbers and G>0.
  • \(A<G<H\), if \(a\) & \(b\) are two distinct -ve real numbers and G<0.
  • Reciprocals of GP form a GP
  • Reciprocals of AP form HP
  • If \(a^x\), \(b^y\), \(c^z\) are in GP then \(x\), \(y\), \(z\) are in HP
  • If \(a^x\), \(b^y\), \(c^z\) are in GP then \(1/x\), \(1/y\), \(1/z\) are in AP
  • The number \(\frac{1}{a}\), \(\frac{a+b}{2ab}\), \(\frac{1}{b}\) are in GP
  • The reciprocal of whose terms form again same type of sequence is HP
  • If the first term of an infinite geometric series is equal to twice of the sum of all the terms of that follows it, then the value of \(r\) is \(1/3\).
  • If \(a\), \(b\), \(c\) form a G.P with common ratio \(r\) \((0<r<1)\).If \(a\),\(2b\),\(3c\) form an A.P then \(r=1/3\).
  • Three numbers of G.P. If we double the middle number we get an A.P the common ratio of G.P is \(2\pm\sqrt{3}\).
  • Every term of GP is the logarithm of each term of AP.
  • Every term of AP is the anti logarithm of term of GP.
  • If \(a\),\(b\),\(c\) are in HP then \(bc\), \(ca\), \(ab\) are in AP.

  • If \(a\),\(b\),\(c\) are in AP then \(\frac{1}{1-a}\), \(\frac{1}{1-b}\), \(\frac{1}{1-c}\) are in HP.

  • The expression $$ \frac{a^n + b^n }{a^{n-1} + b^{n-1}} $$ is:

    • AM when \(n=1\)
    • GM when \(n=1/2\)
    • HM when \(n=0\)
  • The expression $$ \frac{a^{n+1} + b^{n+1} }{an+bn} $$ is:
    • AM when \(n=0\)
    • GM when \(n=-1/2\)
    • HM when \(n=-1\)

Relation with Quadratic Equation

  • If \(\alpha\) & \(\beta\) be roots of the quadratic equation \(x^2-2Ax+G^2=0\) then:

    • AM = \(\frac{\alpha+\beta}{2}\)
    • GM = \(\sqrt{\alpha\beta}\)
    • HM = \(\frac{2\alpha\beta}{\alpha+\beta}\)

  • Let \(a\) & \(b\) are two numbers and if \(a=b\) then $$ \text{GM} = \text{HM} = \text{AM} $$

  • When quadratic equation is \(ax^2+bx+c=0\) then:

    • AM = \(\frac{-b}{2a}\)
    • HM = \(\frac{-2c}{b}\)
    • GM = \(\sqrt{\frac{c}{a}}\)

  • If \(A\) & \(G\) are Arithmetic and Geometric means between \(a\) & \(b\) then the quadratic equation having roots \(a\) & \(b\) will be $$ x^2 - 2Ax + G^2=0 $$

Sigma Notations

\[ \displaystyle\sum^n_{k=1} [k^m-(k-1)^m]=n^m \]
  • Sum \(n\) times unity $$ \displaystyle\sum^n_{k=1} 1=n $$
  • Sum of first \(n\) natural numbers is given by $$ \displaystyle\sum^n_{k=1} k = \frac{n(n+1)}{2} $$
  • The sum of first \(n\) even natural number is given by \(n(n+1)\)
  • The sum of first \(n\) odd natural number is given by \(n^2\)
  • The sum of squares of first \(n\) natural numbers is given by $$ \displaystyle\sum^n_{k=1} k^2=\frac{n(n+1)(2n+1)}{6} $$
  • The sum of squares of first \(n\) odd natural numbers is given by $$ \frac{n(4n^2-1)}{3} $$
  • The sum of squares of first \(n\) even natural number is given by $$ \frac{2n(n+1)(2n+1)}{3} $$
  • The sum of cube of first \(n\) natural number is given by $$ \displaystyle\sum^n_{k=1} k3 = \frac{n(n+1)2}{2} $$
  • The sum of cube of first \(n\) even natural number is given by \(2n^2(n+1)^2\)
  • The sum of cube of first \(n\) odd natural number is given by \(n^2(2n^2-1)\)

nth Term of Repeated Series

  • nth term of series like \(b+\)\(bb+\)\(bbb+\)... is $$ a_n=\frac{b}{9}(10^n-1) $$
  • nth term of series like \(0.b+\)\(0.bb+\)\(0.bb+\)... is $$ a_n=\frac{b}{9}(1-10^n) $$

Vulgar Fraction

A type of decimal fraction in which the decimals repeats endlessly is called recurring or repeated decimals.

For example: 0.31313131...

Every recurring decimal can be converted into a fraction called common fraction or vulgar fraction by using formula for \(S_\infty\).

Conversion

Here are some key points to convert recurring decimals into vulgar fraction within seconds.

Case 1 When all digits are recurring

In this case, in the denominator of vulgar fraction, the number of nines is equal to number of repeating digits and numerator is actually the complete given number without decimal, minus the number before decimal.

Example
  • Vulgar Fraction of \(2.\overline{342}\) will be $$ \frac{2342-2}{999}=\frac{2340}{999} $$
  • \(13.\overline{4235}\) will be $$ \frac{134235-13}{9999}=\frac{13422}{9999} $$
  • \(0.\overline{271}\) will be $$ \frac{0271-0}{999}=\frac{271}{999} $$

Case 2 When all digits are not repeating

In this case, in the denominator of vulgar fraction, the number of nines is equal to the number of repeating digits and after nines we put zeros and the number of zeros is equal to the number of non-repeating digits in decimal part. The numerator is the whole given number without decimal minus the number before repeating digits.

Example
  • \(2.1\overline{341}\) will be $$ \frac{21341-21}{9990}=\frac{21320}{9990} $$
  • \(0.02\overline{1}\) will be $$ \frac{0021-002}{900}=\frac{19}{900} $$

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