Arithmetic and geometric progressions and their sums
A sequence is an ordered list of numbers, and a series is the sum of its terms. In an arithmetic progression each term is found by adding a fixed common difference d, so aₙ = a + (n − 1)d. In a geometric progression each term is found by multiplying by a fixed common ratio r, as in 2, 6, 18, 54, …, where r = 3. The nth term of a GP is arⁿ⁻¹, and the sum of its first n terms is Sₙ = a(rⁿ − 1)/(r − 1) whenever r ≠ 1.
When |r| < 1 the terms of a GP shrink towards zero, and the sum to infinity is a/(1 − r), so 1 + ½ + ¼ + … approaches 2. The arithmetic mean of two numbers a and b is (a + b)/2 and the geometric mean is √(ab). For positive numbers, the arithmetic mean is always at least as large as the geometric mean, with equality only when a = b. Sums of special series, such as 1² + 2² + … + n² = n(n + 1)(2n + 1)/6, also belong to this chapter.
Why it matters
Geometric progressions describe compound interest, population growth, the falling value of a car and radioactive decay, while arithmetic progressions describe steady changes such as fixed yearly salary increments. An infinite GP explains why the recurring decimal 0.333… equals 1/3. The AM–GM inequality is a powerful tool in JEE problems on maximum and minimum values.
Key terms
- Common ratio
- The fixed number each term of a GP is multiplied by to get the next term.
- Sum to infinity
- The value a/(1 − r) that the sum of an infinite GP approaches when |r| < 1.
- Geometric mean
- For two positive numbers a and b, the square root of their product.
- Series
- The sum obtained by adding the terms of a sequence.