Question

If sum of first 9 terms of an $A.P.$ is $81$ and the sum of first $17$ terms is $289$, find its first term and the common difference.

Hint:

When two different partial sums of an A.P. are given, convert each to a linear equation in $a$ and $d$ using $S_n=\frac{n}{2}(2a+(n-1)d)$, then solve the pair.

Answer 1

Step 1: Write the sum formula of an A.P.
For an A.P. with first term $a$ and common difference $d$, \[ S_n=\frac{n}{2}\big(2a+(n-1)d\big). \]

Step 2: Form equations from given sums.
For $n=9$: \[ S_9=\frac{9}{2}(2a+8d)=81 $\Rightarrow$ 9(2a+8d)=162 $\Rightarrow$ 2a+8d=18 $\Rightarrow$ a+4d=9.\tag{1} \] For $n=17$: \[ S_{17}=\frac{17}{2}(2a+16d)=289 $\Rightarrow$ 17(2a+16d)=578 $\Rightarrow$ 2a+16d=34 $\Rightarrow$ a+8d=17.\tag{2} \]

Step 3: Solve (1) and (2).
Subtract (1) from (2): $(a+8d)-(a+4d)=17-9 $\Rightarrow$ 4d=8 $\Rightarrow$ d=2$.
Substitute in (1): $a+4(2)=9 $\Rightarrow$ a+8=9 $\Rightarrow$ a=1$. \boxed{a=1, d=2.}