Sequences and Series

Introduction

Sequence

A sequence is an ordered list of numbers. The various numbers occurring in a sequence are called its terms.

• Denoted by $a_1, a_2, a_3, \dots, a_n, \dots$.
• A sequence containing a finite number of terms is called a finite sequence.
• A sequence is called infinite if it is not a finite sequence.

Series

If $a_1, a_2, a_3, \dots, a_n$ is a sequence, then the expression $a_1 + a_2 + a_3 + \dots + a_n$ is called the series associated with the sequence.

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Arithmetic Progression (A.P.)

A sequence in which the difference between any two consecutive terms is constant.

Standard Form

$a, a+d, a+2d, a+3d, \dots$

• $a$ = first term
• $d$ = common difference

Formulas

1. General Term ($n^{th}$ term): $$a_n = a + (n-1)d$$
2. Sum of first $n$ terms: $$S_n = \frac{n}{2}[2a + (n-1)d]$$ $$S_n = \frac{n}{2}(a + l) \quad \text{(where } l = a_n \text{ is last term)}$$

Arithmetic Mean (A.M.)

If $a, A, b$ are in A.P., then $A$ is the arithmetic mean of $a$ and $b$. $$A = \frac{a+b}{2}$$

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Geometric Progression (G.P.)

A sequence in which the ratio of any term to its preceding term is constant.

Standard Form

$a, ar, ar^2, ar^3, \dots$

• $a$ = first term
• $r$ = common ratio

Formulas

1. General Term ($n^{th}$ term): $$a_n = ar^{n-1}$$
2. Sum of first $n$ terms: $$S_n = \frac{a(r^n - 1)}{r - 1} \quad \text{if } r \neq 1$$
3. Sum of Infinite G.P. ($|r| < 1$): $$S_\infty = \frac{a}{1 - r}$$

Geometric Mean (G.M.)

If $a, G, b$ are in G.P., then $G$ is the geometric mean of $a$ and $b$. $$G = \sqrt{ab}$$

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Relationship Between A.M. and G.M.

For any two positive real numbers $a$ and $b$: $$A.M. \ge G.M.$$ Equality holds if and only if $a = b$.

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Sum of Special Series

1. Sum of first $n$ natural numbers: $$\sum_{k=1}^n k = \frac{n(n+1)}{2}$$
2. Sum of squares of first $n$ natural numbers: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
3. Sum of cubes of first $n$ natural numbers: $$\sum_{k=1}^n k^3 = \left[ \frac{n(n+1)}{2} \right]^2$$

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Common Mistakes

1. Confusing $d$ (difference) with $r$ (ratio).
2. Using G.P. formula for A.P. or vice-versa.
3. Wrong substitution of $n$.
4. Forgetting condition $|r|<1$ for infinite G.P.
5. Writing wrong last term in A.P.

Sequences and Series

Example 1

Find the 10th term of the sequence defined by $a_n = 2n^2 + 1$.

Solution: Put $n=10$: $a_{10} = 2(10)^2 + 1 = 200 + 1 = 201$.

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Example 2

In an A.P., if $a=5$ and $d=-3$, find the 12th term.

Solution: $a_{12} = a + 11d = 5 + 11(-3) = 5 - 33 = -28$.

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Example 3

Which term of the A.P. 3, 8, 13, … is 78?

Solution: $a=3, d=5$. Let $a_n = 78$. $3 + (n-1)5 = 78 \implies 5(n-1) = 75 \implies n-1 = 15 \implies n=16$.

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Example 4

Find the sum of the first 20 terms of the A.P. 1, 4, 7, 10…

Solution: $a=1, d=3, n=20$. $S_{20} = \frac{20}{2}[2(1) + 19(3)] = 10[2 + 57] = 590$.

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Example 5

Insert 3 arithmetic means between 3 and 19.

Solution: Let means be $A_1, A_2, A_3$. Sequence: 3, $A_1, A_2, A_3$, 19. Total terms = 5. $a_5 = 3 + 4d = 19 \implies 4d = 16 \implies d=4$. Means: $3+4=7, 7+4=11, 11+4=15$.

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Example 6

Find the 7th term of the G.P. 2, 4, 8, 16…

Solution: $a=2, r=2$. $a_7 = ar^6 = 2(2)^6 = 2^7 = 128$.

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Example 7

Find the sum of G.P. $1 + \frac{1}{2} + \frac{1}{4} + \dots$ up to 8 terms.

Solution: $a=1, r=1/2$. $S_8 = \frac{1(1 - (1/2)^8)}{1 - 1/2} = \frac{1 - 1/256}{1/2} = 2 \times \frac{255}{256} = \frac{255}{128}$.

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Example 8

Find sum to infinity of G.P. 5, 2.5, 1.25…

Solution: $a=5, r=0.5$. $S_\infty = \frac{a}{1-r} = \frac{5}{1-0.5} = 10$.

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Example 9

Find the geometric mean of 4 and 9.

Solution: $G = \sqrt{4 \times 9} = \sqrt{36} = 6$.

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Example 10

If A.M. and G.M. of two numbers are 10 and 8 respectively, find the numbers.

Solution: $a+b=20, ab=64$. Quadratic: $x^2 - 20x + 64 = 0$. Roots are 16 and 4. Numbers are 4, 16.

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Example 11

Find the sum of first $n$ odd natural numbers.

Solution: A.P. 1, 3, 5… $a=1, d=2$. $S_n = \frac{n}{2}[2 + (n-1)2] = \frac{n}{2}[2n] = n^2$.

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Example 12

Evaluate $\sum_{k=1}^{10} (2k + 3)$.

Solution: $= 2\sum k + \sum 3 = 2\frac{10(11)}{2} + 10(3) = 110 + 30 = 140$.

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Example 13

Which term of G.P. 2, 8, 32 … is 131072?

Solution: $a=2, r=4$. $2 \cdot 4^{n-1} = 131072 \implies 4^{n-1} = 65536 = 4^8$. $n-1 = 8 \implies n=9$.

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Example 14

If sum of 3 numbers in A.P. is 24 and their product is 440, find the numbers.

Solution: Let numbers be $a-d, a, a+d$. Sum $= 3a = 24 \implies a=8$. Product: $8(64-d^2) = 440 \implies 64-d^2 = 55 \implies d^2 = 9 \implies d=\pm 3$. Numbers: 5, 8, 11.

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Example 15

Calculate $1^2 + 2^2 + \dots + 10^2$.

Solution: Formula: $\frac{n(n+1)(2n+1)}{6}$. For $n=10$: $\frac{10(11)(21)}{6} = 385$.

Questions and Answers (Board Exam)

Q1. Define Arithmetic Progression.

Answer: A sequence where the difference between consecutive terms is constant is called an Arithmetic Progression.

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Q2. What is the relation between A.M. and G.M.?

Answer: For any two positive real numbers, Arithmetic Mean $\ge$ Geometric Mean ($A \ge G$).

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Q3. Write the formula for sum of infinite G.P.

Answer: $S_\infty = \frac{a}{1-r}$, valid only when $|r| < 1$.

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Q4. What is the sum of first $n$ natural numbers?

Answer: $\frac{n(n+1)}{2}$.

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Q5. If $a, b, c$ are in G.P., write the condition.

Answer: $b^2 = ac$ or $\frac{b}{a} = \frac{c}{b}$.