Sequences & Series

CHAPTER 4

4.1

Arithmetic Sequences (A.P.)

Name	Formula	Note
General Term	$T(n) = a + (n-1)d$	$a$ = first term, $d$ = common difference
Sum Formula 1	$S(n) = \dfrac{n}{2}[2a + (n-1)d]$	When $a$, $d$, $n$ known
Sum Formula 2	$S(n) = \dfrac{n(a + l)}{2}$	$l$ = last term

Example 1

$a = 2$, $d = 3$

$T(20) = 2 + 19 \times 3 = 59$

$S(20) = \dfrac{20(2 + 59)}{2} = 610$

4.2

Geometric Sequences (G.P.)

Name	Formula	Note
General Term	$T(n) = ar^{n-1}$	$a$ = first term, $r$ = common ratio
Sum Formula	$S(n) = \dfrac{a(1-r^n)}{1-r}$	$r \neq 1$

4.3

Infinite Geometric Series

Sum to Infinity

$S_\infty = \dfrac{a}{1-r}$

Condition: $|r| < 1$

$0.\dot{3} = \dfrac{0.3}{1-0.1} = \dfrac{0.3}{0.9} = \dfrac{1}{3}$

DSE 2021 Q17

A G.P. has first term 4 and common ratio 3. Find the sum of first 5 terms.

A. 484 B. 364 C. 324 D. 244

$S(5) = \dfrac{4(3^5-1)}{3-1} = \dfrac{4 \times 242}{2} = 484$

Answer: A

1. A.P.: 3, 7, 11, ... Find the 15th term.

2. G.P. with $a = 2$, $r = \frac{1}{2}$. Find sum of first 6 terms.

3. Find the sum of the infinite series $1 + \frac{1}{3} + \frac{1}{9} + ...$

1. 59 2. $\frac{63}{16}$ 3. $\frac{3}{2}$

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