Sequences, Series, and Variation: JAMB Mathematics Notes

Patterns in number sequences (arithmetic and geometric progressions) and how one quantity varies with another.

Arithmetic progression (AP)

nth term: $T_{n} = a + (n - 1) d$ , where $a$ is the first term and $d$ is the common difference.

Sum of n terms: $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .

Example 1: Find the 10th term of the AP $3, 7, 11, 15, \dots$

Here $a = 3, d = 4$ . $T_{10} = 3 + (10 - 1) (4) = 3 + 36 = 39$ .

Example 2: Find the sum of the first 15 terms of the same AP.

$S_{15} = \frac{15}{2} [2 (3) + (15 - 1) (4)] = 7.5 [6 + 56] = 7.5 \times 62 = 465$ .

nth term: $T_{n} = a r^{n - 1}$ , where $r$ is the common ratio.

Sum of n terms: $S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1}$ (for $r \neq = 1$ ).

Sum to infinity (only when $- 1 < r < 1$ ): $S_{\infty} = \frac{a}{1 - r}$ .

Example 3: Find the 6th term of the GP $2, 6, 18, 54, \dots$

Here $a = 2, r = 3$ . $T_{6} = 2 \times 3^{6 - 1} = 2 \times 243 = 486$ .

Example 4: Find the sum to infinity of $8, 4, 2, 1, \dots$

Here $a = 8, r = \frac{1}{2}$ . $S_{\infty} = \frac{8}{1 - 0.5} = \frac{8}{0.5} = 16$ .

Example 5: $y$ varies directly as $x$ . When $x = 5$ , $y = 20$ . Find $y$ when $x = 8$ .

Find $k$ : $20 = k (5)$ , so $k = 4$ . Then $y = 4 (8) = 32$ .