The common difference of the A.P.: a_1,a_2, ,a_m is 13 more than the common difference of …

Question

The common difference of the A.P.: ${a}_{1},{a}_{2},\ldots ,{a}_{m}$ is 13 more than the common difference of the A.P.: ${b}_{1},{b}_{2},\ldots ,{b}_{n}$. If ${b}_{31}=-277,{b}_{43}=-385$ and ${a}_{78}=327$, then ${a}_{1}$ is equal to

Accepted Answer

Find common difference of second A.P.

It is given that ${b}_{31}=-277$, ${b}_{43}=-385$

Let ${d}_{b}$ is common difference of second A.P.

$\begin{gathered} {b}_{31}={b}_{1}+30{d}_{b}---(1) \ {b}_{43}={b}_{1}+42{d}_{b}---(2) \end{gathered}$

Subtract (1) from (2)

$\begin{gathered} {b}_{43}-{b}_{31}=(42-30){d}_{b} \ -385-(-277)=12{d}_{b} \ {d}_{b}=\frac{-108}{12}=-9 \end{gathered}$

Find the common difference ( ${d}_{a}$ ) of the first A.P.

The problem states that ${d}_{a}$ is 13 more than ${d}_{b}$ :

$\begin{gathered} {d}_{a}={d}_{b}+13 \ {d}_{a}=-9+13=4 \end{gathered}$

Calculate ${a}_{1}$

We are given ${a}_{78}=327$. Using the formula ${a}_{n}={a}_{1}+(n-1){d}_{a}$ :

$\begin{gathered} 327={a}_{1}+(78-1)(4) \ 327={a}_{1}+(77)(4) \ 327={a}_{1}+308 \ {a}_{1}=327-308=19 \end{gathered}$

Solutions · 1