Question

Find four consecutive terms in an A.P. whose sum is 36 and the product of the 2nd and 4th term is 105. (The terms are in ascending order.) Right answer will be marks as brainliest!!

Answer 1

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

Given that, The terms of an AP are in increasing order.

So, it means, common difference > 0.

Let assume that four numbers in AP as

$\begin{gathered}\begin{gathered}\bf\: 4 \: numbers \: in \: AP \: are-\begin{cases} &\sf{a - 3d} \\ &\sf{a - d} \\ &\sf{a + d} \\ &\sf{a + 3d} \end{cases}\end{gathered}\end{gathered}$

According to statement, sum of four consecutive terms of AP is 36.

$\rm \: a - 3d + a - d + a + d + a + 3d = 36$

$\rm \: 4a = 36$

$\rm\implies \:\boxed{ \rm{ \:a = 9 \: }} \: \cdots \cdots(1)$

According to second condition, the product of 2nd and 4th term is 105.

$\rm \: (a - d)(a + 3d) = 105$

$\rm \: (9 - d)(9 + 3d) = 105 \: \: \: \: \: \: \: \{using \: (1) \}$

$\rm \: 81 + 27d - 9d - {3d}^{2} = 105$

$\rm \: 81 + 18d - {3d}^{2} = 105$

$\rm \: {3d}^{2} - 18d + 105 - 81 = 0$

$\rm \: {3d}^{2} - 18d + 24 = 0$

$\rm \: 3({d}^{2} - 6d + 8) = 0$

$\rm \: {d}^{2} - 6d + 8 = 0$

$\rm \: {d}^{2} - 4d - 2d + 8 = 0$

$\rm \: d(d - 4) - 2(d - 4) = 0$

$\rm \: (d - 4)(d - 2) = 0$

$\bf\implies \:d \: = \: 4 \: \: \: or \: \: \: d = 2$

So, two cases aries :-

Case :- 1 When a = 9 and d = 2

$\begin{gathered}\begin{gathered}\bf\: 4 \: numbers \: in \: AP \: are-\begin{cases} &\sf{a - 3d = 9 - 6 = 3} \\ &\sf{a - d = 9 - 2 = 7} \\ &\sf{a + d = 9 + 2 = 11} \\ &\sf{a + 3d = 9 + 6 = 15} \end{cases}\end{gathered}\end{gathered}$

Case :- 2 When a = 9 and d = 4

$\begin{gathered}\begin{gathered}\bf\: 4 \: numbers \: in \: AP \: are-\begin{cases} &\sf{a - 3d = 9 - 12 = - 3} \\ &\sf{a - d = 9 - 4 = 5} \\ &\sf{a + d = 9 + 4 = 13} \\ &\sf{a + 3d = 9 + 12 = 21} \end{cases}\end{gathered}\end{gathered}$

$\rule{190pt}{2pt}$

Additional Information :-

↝ nᵗʰ term of an arithmetic progression is,

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

↝ Sum of n  terms of an arithmetic progression 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.

• aₙ is the nᵗʰ term.

• a is the first term of the progression.

• n is the no. of terms.

• d is the common difference.