# If fourth term of an A. P. is zero, show that its 25th term is thrice its 11th term.? ( A.P. = arithmetic progression )

## Proof

Feb 19, 2016

(see below)

#### Explanation:

For an arithmetic progression with a difference of $d$ between terms and an initial value of ${a}_{0}$:
$\textcolor{w h i t e}{\text{XXX}} {a}_{k} = {a}_{0} + k \cdot d$

We are told that ${a}_{4} = {a}_{0} + 4 d = 0$

${a}_{11} = {a}_{0} + 11 d$
$\textcolor{w h i t e}{\text{XXX}} = {a}_{0} + 4 d + 7 d$
$\textcolor{w h i t e}{\text{XXX}} = {a}_{4} + 7 d$
$\textcolor{w h i t e}{\text{XXX}} = 7 d$

${a}_{25} = {a}_{0} + 25 d$
$\textcolor{w h i t e}{\text{XXX}} = {a}_{0} + 4 d + 21 d$
$\textcolor{w h i t e}{\text{XXX}} = {a}_{4} + 21 d$
$\textcolor{w h i t e}{\text{XXX}} = 21 d$

${a}_{25} = 3 \times {a}_{11}$