Question

There are four terms in an A.P. In which sum of the first and last term is 13 and the <br /><br />product of middle terms is 40. Then the terms of the given A.P​

Answer 1

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

Given that,

Four terms are in AP.

Let assume that four terms in

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:AP \: are-\begin{cases} &\sf{a - 3d} \\ &\sf{a - d}\\ &\sf{a + d}\\ &\sf{a + 3d} \end{cases}\end{gathered}\end{gathered}$

Further given that,

Sum of first and last term is 13.

$\rm :\longmapsto\:a - 3d + a + 3d = 13$

$\rm :\longmapsto\:2a = 13$

$\bf\implies \:a = \dfrac{13}{2} - - - (1)$

Further given that,

Product of middle terms is 40

$\rm :\longmapsto\:(a - d)(a + d) = 40$

$\rm :\longmapsto\: {a}^{2} - {d}^{2} = 40$

$\rm :\longmapsto\: {\bigg(\dfrac{13}{2} \bigg) }^{2} - {d}^{2} = 40$

$\rm :\longmapsto\:\dfrac{169}{4} - {d}^{2} = 40$

$\rm :\longmapsto\:- {d}^{2} = 40 - \dfrac{169}{4}$

$\rm :\longmapsto\:- {d}^{2} = \dfrac{160 - 169}{4}$

$\rm :\longmapsto\:- {d}^{2} = \dfrac{ - 9}{4}$

$\rm :\longmapsto\:{d}^{2} = \dfrac{9}{4}$

$\bf\implies \:d \: = \: \pm \: \dfrac{3}{2}$

So, two cases arises,

Case :- 1

$\rm :\longmapsto\:When \: a = \dfrac{13}{2} \: and \:d = \dfrac{3}{2}$

So, numbers in

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:AP \: are-\begin{cases} &\sf{a - 3d = 2} \\ &\sf{a - d = 5}\\ &\sf{a + d = 8}\\ &\sf{a + 3d = 11} \end{cases}\end{gathered}\end{gathered}$

Case :- 2

$\rm :\longmapsto\:When \: a = \dfrac{13}{2} \: and \: d = \: - \: \dfrac{3}{2}$

So, numbers in

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:AP \: are-\begin{cases} &\sf{a - 3d = 11} \\ &\sf{a - d = 8}\\ &\sf{a + d = 5}\\ &\sf{a + 3d = 2} \end{cases}\end{gathered}\end{gathered}$

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.

Answer 2

Answers:-

11, 8, 5, 2

Hope it helps you