Question

If the $6^{th}$ term of an A.P is equal to four times its first term and the sum of first six term is $75$, find the first term and the common difference?

Answer 1

Step 1. Form equations based on given information about Arithmetic Progression

For given A.P.

Let the first term $=a$

Common difference $=d$

Number of terms $=n$

$\therefore n^{th}$ term of A.P. is $\left(a_n\right)$$=a+(n-1)d$

The $6^{th}$ term $=a+(6-1)d$

$=a+5d$

The given condition is, the $6^{th}$ term of an A.P is equal to four times its first term

$$\begin{gathered} \therefore a+5d=4a \\ \Rightarrow 3a-5d=0...............................\left(1\right) \end{gathered}$$

Another given condition is, the sum of first six term is $75$.

We know that sum of $n$ terms of an A.P is

${\mathrm{S}}_{\mathrm{n}}=a·n+\frac{n\left(n-1\right)}{2}·d$

$\therefore 6a+15d=75.....\left(2\right)$

Step 2. Find the first term and common difference.

Using equation $\left(1\right)$ and $\left(2\right)$ we get,

$$\begin{gathered} 6a+15d=75 \\ \Rightarrow 6a+9a=75\because \left[3a=5d\right] \\ \Rightarrow 15a=75 \\ \Rightarrow a=5 \end{gathered}$$

Therefore, the first term is $5$

Putting the value of the first term in equation $\left(1\right)$ we get,

$$\begin{gathered} 5d=3a \\ \Rightarrow 5d=3\times 5 \\ \Rightarrow d=3 \end{gathered}$$

The common difference is $3$

Hence, the first term is $5$ and the common difference is $3$.