'Sequences and Series'

Sequences and Series

A sequence is an ordered list of numbers following a particular rule. Each number is called a term. A series is the sum of the terms of a sequence.

Notation: a_1, a_2, a_3, ..., a_n denotes a sequence with a_n as the nth term.

Arithmetic Progression (AP)

A sequence where the difference between consecutive terms is constant. This constant difference is called the common difference d.

General Form

a, a+d, a+2d, a+3d, ...

nth Term

a_n = a + (n-1)d

Sum of n Terms

S_n = n/2 [2a + (n-1)d] S_n = n/2 [a + l] where l = last term

Arithmetic Mean

If three numbers are in AP: a-d, a, a+d Arithmetic mean of a and b is (a+b)/2.

Selection of Terms in AP

• 3 terms: a-d, a, a+d
• 4 terms: a-3d, a-d, a+d, a+3d
• 5 terms: a-2d, a-d, a, a+d, a+2d

Geometric Progression (GP)

A sequence where the ratio of consecutive terms is constant. This ratio is called the common ratio r.

General Form

a, ar, ar^2, ar^3, ...

nth Term

a_n = ar^(n-1)

Sum of n Terms

S_n = a(r^n - 1)/(r - 1) for r != 1 S_n = na for r = 1

Sum of Infinite GP

S_infinity = a/(1 - r) when |r| < 1

Geometric Mean

If three numbers are in GP: a/r, a, ar Geometric mean of a and b is sqrt(ab).

Selection of Terms in GP

• 3 terms: a/r, a, ar
• 4 terms: a/r^3, a/r, ar, ar^3
• 5 terms: a/r^2, a/r, a, ar, ar^2

Harmonic Progression (HP)

A sequence a_1, a_2, a_3, ... is in HP if the reciprocals 1/a_1, 1/a_2, 1/a_3, ... are in AP.

nth Term of HP

If a is the first term of HP, the nth term = 1/(1/a + (n-1)d) where d is the common difference of the corresponding AP.

Harmonic Mean

Harmonic mean of a and b is 2ab/(a+b).

Relationship Between AM, GM, HM

• AM >= GM >= HM
• (AM)(HM) = (GM)^2

Arithmetic-Geometric Series (AGP)

A series of the form a + (a+d)r + (a+2d)r^2 + ...

Sum of n Terms of AGP

Let S = a + (a+d)r + (a+2d)r^2 + ... + (a+(n-1)d)r^(n-1) Multiply by r and subtract to find S_n.

Sum of Infinite AGP

When |r| < 1: S_infinity = a/(1-r) + dr/(1-r)^2

Method of Differences

When the differences between consecutive terms form an AP or GP, use this method to find the nth term and sum.

Special Sums

Sum of First n Natural Numbers

sum_(k=1)^n k = 1 + 2 + 3 + ... + n = n(n+1)/2

Sum of Squares of First n Natural Numbers

sum_(k=1)^n k^2 = 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6

Sum of Cubes of First n Natural Numbers

sum_(k=1)^n k^3 = 1^3 + 2^3 + ... + n^3 = [n(n+1)/2]^2

Proof for sum of squares: Using identity (k+1)^3 - k^3 = 3k^2 + 3k + 1 and summing from k=1 to n.

Worked Examples

Example 1: Find the 10th term of AP: 3, 7, 11, 15, ... Solution: a = 3, d = 4. a_10 = 3 + 9*4 = 39.

Example 2: How many terms of AP: 24, 21, 18, ... amount to 78? Solution: a=24, d=-3, S_n=78. n/2[48 + (n-1)(-3)] = 78. n(51-3n) = 156. 3n^2 - 51n + 156 = 0. n=4 or n=13. Both are valid.

Example 3: Find the sum of the infinite GP: 1 + 1/2 + 1/4 + 1/8 + ... Solution: a=1, r=1/2. S = 1/(1-1/2) = 2.

Example 4: Find sum to n terms: 1^2 + 2^2 + 3^2 + ... + n^2 = 10. Solution: Using the formula, n(n+1)(2n+1)/6 = 10. Since n must be a natural number, n=3 (since 3*4*7/6 = 14) - but 14 > 10. None satisfies exactly for integer n. The sum to 3 terms is 14 which exceeds 10, so n=2, sum = 5, or n=3, sum = 14. So n=2 if sum must be <=10.

Common Mistakes

1. r in GP: For |r| < 1, any term T_n = ar^(n-1) where r is the factor, not the ratio written in reverse.
2. HP confusion: Always convert HP to AP first before solving.
3. Sigma notation: sum(ab) != (sum a)(sum b).
4. Infinite GP formula only valid for |r|<1: Using the infinite sum formula when |r| >= 1 gives wrong results.

ISC Exam Focus

• Theory (70%): AP and GP formulas, AM, GM, HM definitions and relationships.
• Application (30%): Word problems, special sums, AGP sums.
• Key ISC question types: "Find the sum of n terms of series ...", "Three numbers in AP/GP sum to ... and product to ...".
• Special sums Sigma n, Sigma n^2, Sigma n^3 are frequently tested.

Self-Test Questions

Q1: Find the 15th term of AP: 5, 9, 13, 17, ... Answer: a=5, d=4. a_15 = 5 + 14*4 = 61.

Q2: Find the sum of the first 20 terms of AP: 2, 5, 8, ... Answer: a=2, d=3. S_20 = 20/2[4 + 19*3] = 10[61] = 610.

Q3: Find 3 numbers in GP whose sum is 13 and product is 27. Answer: Let terms be a/r, a, ar. Product = a^3 = 27 => a = 3. Sum = 3/r + 3 + 3r = 13 => 3/r + 3r = 10 => 3r^2 - 10r + 3 = 0 => r = 3 or 1/3. Terms: 1, 3, 9 or 9, 3, 1.

Q4: Find the sum to infinity: 1 + 1/3 + 1/9 + 1/27 + ... Answer: a=1, r=1/3. S = 1/(1-1/3) = 3/2.

Q5: Find Sigma(k=1 to 10) k^2. Answer: 10*11*21/6 = 2310/6 = 385.

Q6: Insert 3 arithmetic means between 3 and 19. Answer: Let means be A_1, A_2, A_3. Total terms = 5. a=3, l=19. 19 = 3 + 4d => d = 4. Means: 7, 11, 15.