Question

Find the 3 consecutive terms which are in A P. Whose sum is 27 and sum of their squares is
293.​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that three consecutive terms in AP series be

$\begin{gathered}\begin{gathered}\bf\: \begin{cases} &\sf{x - y} \\ \\ &\sf{x}\\ \\ &\sf{x + y} \end{cases}\end{gathered}\end{gathered}$

So, According to statement

Sum of three terms of AP is 27

$\rm \: x - y + x + x + y = 27$

$\rm \: 3x = 27$

$\rm\implies \:\boxed{\tt{ \: x \: = \: 9 \: }}$

According to statement again,

Sum of the squares of 3 terms is 293

$\rm \: {(x - y)}^{2} + {x}^{2} + {(x + y)}^{2} = 293$

$\rm \: {x}^{2} + {y}^{2} - 2xy + {x}^{2} + {x}^{2} + {y}^{2} + 2xy = 293$

$\rm \: {3x}^{2} + 2 {y}^{2} = 293$

On substituting the value of x = 9, we get

$\rm \: {3(9)}^{2} + 2 {y}^{2} = 293$

$\rm \: 243 + 2 {y}^{2} = 293$

$\rm \: 2 {y}^{2} = 293 - 243$

$\rm \: 2 {y}^{2} = 50$

$\rm \: {y}^{2} = 25$

$\rm\implies \:\boxed{\tt{ \: y \: = \: \pm \: 5}}$

So, Two cases arises.

Case :- 1

When x = 9 and y = 5

So, three consecutive terms are

$\begin{gathered}\begin{gathered}\bf\: \begin{cases} &\sf{x - y = 9 - 5 = 4} \\ \\ &\sf{x = 9}\\ \\ &\sf{x + y = 9 + 5 = 14} \end{cases}\end{gathered}\end{gathered}$

Case :- 2

When x = 9 and y = - 5

So, three consecutive terms are

$\begin{gathered}\begin{gathered}\bf\: \begin{cases} &\sf{x - y = 9 + 5 = 14} \\ \\ &\sf{x = 9}\\ \\ &\sf{x + y = 9 - 5 = 4} \end{cases}\end{gathered}\end{gathered}$

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ADDITIONAL INFORMATION

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

aₙ is the nᵗʰ term.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of AP.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.