A sequence is an ordered list of numbers following a definite pattern. Each number in the list is called a term. A series is the sum of the terms of a sequence. Understanding sequences and series is essential for many areas of mathematics, including calculus and discrete mathematics.
- Types of Sequences:
- Arithmetic Progression (AP): Each term is obtained by adding a fixed number (called the common difference, d) to the previous term.
- Geometric Progression (GP): Each term is obtained by multiplying the previous term by a fixed number (called the common ratio, r).
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Key Formulas
- Arithmetic Progression:
- General term: an = a + (n - 1)d, where a = first term, d = common difference
- Sum of n terms: Sn = n/2 × [2a + (n - 1)d] or Sn = n/2 × (a + l), where l = last term
- Geometric Progression:
- General term: an = a × rn-1, where a = first term, r = common ratio
- Sum of n terms: Sn = a(rn - 1)/(r - 1) when r ≠ 1; Sn = na when r = 1
- Sum of infinite GP (|r| < 1): S = a/(1 - r)
- Special Sums:
- Sum of first n natural numbers: 1 + 2 + 3 + ... + n = n(n+1)/2
- Sum of squares: 12 + 22 + ... + n2 = n(n+1)(2n+1)/6
- Sum of cubes: 13 + 23 + ... + n3 = [n(n+1)/2]2
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Worked Examples
Find the 20th term of the AP: 3, 7, 11, 15, ...
Here a = 3, d = 7 - 3 = 4, n = 20.
a20 = 3 + (20 - 1) × 4 = 3 + 76 = 79
Find the sum of first 15 terms of the AP: 5, 8, 11, ...
a = 5, d = 3, n = 15.
S15 = 15/2 × [2(5) + 14(3)] = 15/2 × [10 + 42] = 15/2 × 52 = 390
The 3rd and 6th terms of a GP are 12 and 96. Find the first term and common ratio.
a3 = ar2 = 12 and a6 = ar5 = 96.
Dividing: r3 = 96/12 = 8, so r = 2. Then a(4) = 12, a = 3. First term = 3, r = 2
Find the sum of the infinite GP: 1 + 1/3 + 1/9 + ...
a = 1, r = 1/3. Since |r| < 1, S = 1/(1 - 1/3) = 1/(2/3) = 3/2
Find the sum: 12 + 22 + 32 + ... + 102.
Using formula: n(n+1)(2n+1)/6 with n = 10:
= 10 × 11 × 21 / 6 = 2310/6 = 385
If a, b, c are in AP and a, b, c are also in GP, find the values.
In AP: b - a = c - b, so 2b = a + c.
In GP: b2 = ac.
From AP: a + c = 2b. From GP: ac = b2.
This means a = b = c (constant sequence) is the simplest solution.
Find three numbers in AP whose sum is 15 and product is 80.
Let numbers be a - d, a, a + d. Sum = 3a = 15, so a = 5.
Product = (5 - d)(5)(5 + d) = 5(25 - d2) = 80. So 25 - d2 = 16, d2 = 9, d = ±3.
Numbers are 2, 5, 8 or 8, 5, 2.
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Common mistakes
- Confusing the index: an uses n starting at 1, not 0. Always verify a1 gives the first term.
- In GP, when r is negative, alternate terms change sign — keep track of signs carefully.
- The formula S = a/(1-r) for infinite GP only applies when |r| < 1. If |r| ≥ 1, the series diverges.
Summary
APs have constant differences; GPs have constant ratios. Master the general term and sum formulas. The special sum formulas for natural numbers, squares, and cubes are frequently tested.