An Arithmetic Progression is a sequence where each term is obtained by adding a fixed number (called the common difference) to the previous term.

Key Terms

• First term ($a_1$): The starting number
• Common difference ($d$): The amount added each time to get the next term
• $n$th term ($a_n$): The term in position $n$

The $n$th Term of an AP

$a_n = a_1 + (n - 1)d$

Example

Consider the sequence: 5, 8, 11, 14, 17, ...

• First term ($a_1$) = 5
• Common difference ($d$) = 3 (because 8 − 5 = 3)

To find the 10th term: $a_{10} = 5 + (10 - 1) \times 3 = 5 + 27 = 32$

A Geometric Progression is a sequence where each term is obtained by multiplying the previous term by a fixed number (called the common ratio).

Key Terms

• First term ($a_1$): The starting number
• Common ratio ($r$): The multiplier used to get the next term
• $n$th term ($a_n$): The term in position $n$

The $n$th Term of a GP

$a_n = a_1 \times r^{(n-1)}$

Example

Consider the sequence: 3, 6, 12, 24, 48, ...

• First term ($a_1$) = 3
• Common ratio ($r$) = 2 (because 6 ÷ 3 = 2)

To find the 7th term: $a_7 = 3 \times 2^{(7-1)} = 3 \times 2^6 = 3 \times 64 = 192$

When we add the terms of a progression, we get a series.

Sum of an Arithmetic Progression

$S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)$

Or equivalently: $S_n = \frac{n}{2} \times (a_1 + a_n)$

Example

Find the sum of the first 8 terms of: 2 + 5 + 8 + 11 + ...

• $a_1 = 2$, $d = 3$, $n = 8$

$S_8 = \frac{8}{2} \times (2 \times 2 + (8 - 1) \times 3)$ $S_8 = 4 \times (4 + 21) = 4 \times 25 = 100$

Sum of a Geometric Progression

$S_n = a_1 \times \frac{r^n - 1}{r - 1} \quad \text{(when } r \neq 1\text{)}$

Example

Find the sum of the first 6 terms of: 4 + 12 + 36 + 108 + ...

• $a_1 = 4$, $r = 3$, $n = 6$

$S_6 = 4 \times \frac{3^6 - 1}{3 - 1} = 4 \times \frac{729 - 1}{2} = 4 \times \frac{728}{2} = 4 \times 364 = 1456$

When three numbers form an AP (like $a, b, c$), the middle term $b$ is the arithmetic mean of $a$ and $c$: $b = \frac{a + c}{2}$

When three numbers form a GP, the middle term $b$ is the geometric mean of $a$ and $c$: $b = \sqrt{ac}$

In Tanzania, sequences and series appear in many daily situations. For example, when saving money in a savings group (* chama *), a member may contribute a fixed amount each week — this forms an arithmetic progression. If you deposit 50,000 TZS in the first week, 55,000 TZS in the second week, and increase each weekly contribution by 5,000 TZS, you are working with an AP. Knowing the sum formula helps you calculate the total savings after a full year (52 weeks) to plan for school fees or household needs.