Question

$\begin{array}{c}\text{The fourth term of an A.P. is}11.\text{The sum of the fifth and seventh terms of the A.P. is}34.\text{Find}\\ \text{its common difference.}\\ \text{Find the sum of the first}25\text{terms of an AP whose nth term is given by an}=7-3n\end{array}$

Answer 1

Given,

$a_4=a+3d=11$......(1)

$a_5+a_7=a+4d+a+6d=34$

$\Rightarrow 2a+10d=34$

$\therefore a+5d=17$.......(2)

(2)-(1) gives,

$d=3$

Given,

$a_n=7-3n$

$a_1=7-3=4$

$a_2=7-3\left(2\right)=1$

$\therefore d=a_2-a_1=1-4=-3$

Formula,

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

$=\frac{25}{2}\left[2\right(4)+(25-1\left)\right(-3\left)\right]$

$=\frac{25}{2}[8-72]$

$=-800$