Question

Let $a_1,a_2,a_3,..$. be an increasing sequence of natural numbers, which are in an arithmetic progression with common difference d. Suppose $a_1+a_2+a_3=27$ and $a_1^2+a_2^2+a_3^2=275$. Then the value of $a_1,d$ are

• A. $a_1=3,d=2$
• B. $a_1=4,d=5$
• C. $a_1=5,d=4$
• D. $a_1=2,d=3$
• E. $a_1=8,d=1$

Answer 1

Answer: (C)

$a_1=5,d=4$

Answer 2

Given that,
$a_1,a_2,a_3…$ are in A.P
$a_1+a_2+a_3=27$ -------(1)
$a_1^2+a_2^2+a_3^2=275$ -------(2)
We know that,
$a_2=a_1+d$
$a_3=a_1+2d$
from the equation (1)
$a_1+(a_1+d)+(a_1+2d)=27$
$⇒a_1+d_1=3$ -----(3) (Means $a_2=9$)
from equation (2)
$a_1^2+a_2^2+a_3^2=275$
$⇒a_1^2+a_3^2=275-81$
$⇒a_1^2+a_3^2=194$ -------(4)
So , Let us test from option to find the value of $a_1$ and $d$
From options mentioned, it can clearly be said that the value must be 4 and 5 with respect to equation (1)
let us check by taking $a_1=5 \text{ and } d=4$ that satisfies the equation 2 or not
Hence , from equation (4) we found that the assumed values are respectively correct as LHS =RHS.

So, the correct option is (C): $a_1=5,d=4$