Sum of a Geometric Progression (GP): Formula and Examples

The sum of a geometric progression totals terms that grow or shrink by a constant factor. For a GP a, ar, ar^2, …, ar^(n-1) with ratio r not equal to 1, the sum of the first n terms is S_n = a times (r^n - 1)/(r - 1). It comes from a neat trick: multiply the whole sum by r and subtract, so every middle term cancels.

GP sums matter because compounding processes accumulate fast. The total distance covered by a bouncing ball, the future value of regular investments earning compound interest, and the number of nodes in a complete binary tree are all geometric sums. Without the formula, adding dozens of rapidly growing terms by hand would be hopeless.

Key Concepts

• S_n: the sum of the first n terms of the GP.
• Derivation idea: multiply S_n by r and subtract from S_n. Every middle term cancels, leaving (1 - r) times S_n = a times (1 - r^n).
• Sign-friendly form: S_n = a times (1 - r^n)/(1 - r) is algebraically identical and avoids a negative denominator when r is below 1.
• The r = 1 case: the sequence is constant, a, a, a, …, so S_n = n times a. The main formula cannot be used because its denominator becomes zero.

Formulas

• Sum (r not equal to 1): S_n = a times (r^n - 1)/(r - 1) = a times (1 - r^n)/(1 - r).
• Sum (r = 1): S_n = n times a.
• Assumptions: n is a positive integer; a is non-zero; r is constant.

Worked Example

Add the first 10 terms of 2 + 4 + 8 + … . Here a = 2 and r = 2, so S_10 = 2 times (2^10 - 1)/(2 - 1) = 2 times 1023 = 2046. Check: 2 + 4 + 8 + 16 = 30, and the formula gives 2 times 15 = 30. For a shrinking case, a = 100, r = 0.5, n = 5: S_5 = 100 times (1 - 0.5^5)/0.5 = 193.75, matching 100 + 50 + 25 + 12.5 + 6.25.

Common Misconceptions

• The formula breaks when r is below 1: it does not, but the denominator (r - 1) turns negative. Switch to the form a times (1 - r^n)/(1 - r) so every quantity stays positive and the arithmetic is comfortable.
• Using the formula when r = 1: dividing by (r - 1) means dividing by zero. A constant sequence like 7, 7, 7, … simply sums to n times 7.
• S_n = a times r^n/(r - 1): dropping the -1 in the numerator is a common error; without it the sum overshoots by a/(r - 1).

Real-World Applications

• Bouncing ball: a ball dropped from height h with bounce ratio r travels h times (1 + r)/(1 - r) in total vertical distance, a geometric sum.
• Loans: amortisation schedules and the total repaid on a compound-interest loan are geometric sums.
• Computing: a complete binary tree of height n has 1 + 2 + 4 + … + 2^n = 2^(n+1) - 1 nodes.

The sum of a GP packs n compounding terms into S_n = a times (r^n - 1)/(r - 1), derived by the multiply-and-subtract cancellation. Choose the (1 - r^n)/(1 - r) arrangement when r is a fraction, handle r = 1 as the special constant case, and never drop the -1 from the numerator.