Find the general term for AP and GP and use them to derive formulae for the sums of APs and GPs

A geometric progression is a sequence where each term is found by multiplying the previous term by a constant called the common ratio (r). The first term is denoted by a.

General Term of a GP

The general (nth) term of a GP is given by:

$a_n = a \cdot r^{n-1}$

This formula works because:

• First term: a₁ = a × r⁰ = a
• Second term: a₂ = a × r¹ = ar
• Third term: a₃ = a × r² = ar²
• And so on...

Deriving the Sum Formula for a GP

To derive the sum formula, start with the series:

$S_n = a + ar + ar^2 + ... + ar^{n-1}$

Multiply the entire series by r:

$rS_n = ar + ar^2 + ar^3 + ... + ar^n$

Subtract the second equation from the first:

$S_n - rS_n = (a + ar + ar^2 + ... + ar^{n-1}) - (ar + ar^2 + ... + ar^n)$

$S_n(1 - r) = a - ar^n$

$S_n = a \cdot \frac{1 - r^n}{1 - r}$ (for r ≠ 1)

This is the sum formula for a Geometric Progression.

Worked Example

Find the sum of the first 7 terms of the GP: 3, 6, 12, 24, ...

Solution:

Given: first term a = 3, common ratio r = 6 ÷ 3 = 2, number of terms n = 7

Using the sum formula:

$S_7 = 3 \cdot \frac{1 - 2^7}{1 - 2}$

$S_7 = 3 \cdot \frac{1 - 128}{-1}$

$S_7 = 3 \cdot \frac{-127}{-1}$

$S_7 = 3 \times 127 = 381$

Therefore, the sum of the first 7 terms is 381.

In Tanzania, arithmetic and geometric progressions are used in financial planning. For example, if a farmer saves money in a mobile money account that adds a fixed amount (like 5,000 TZS) each month, this follows an arithmetic progression. If instead the farmer invests money in a business that earns a fixed percentage interest (like 10% per month), this follows a geometric progression. Knowing how to calculate these sums helps predict total savings or earnings over time, enabling better budgeting and decision-making for small businesses and families across Tanzania.