Verified answer

[tex]\large\underline{\sf{Solution-}}[/tex]

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

[tex]\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}[/tex]

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

[tex]\rm \: a - 3d + a - d + a + d + a + 3d = 36 \\ [/tex]

[tex]\rm \: 4a = 36 \\ [/tex]

[tex]\rm\implies \:\boxed{ \rm{ \:a = 9 \: }} \: \cdots \cdots(1) \\ [/tex]

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

[tex]\rm \: (a - d)(a + 3d) = 105 \\ [/tex]

[tex]\rm \: (9 - d)(9 + 3d) = 105 \: \: \: \: \: \: \: \{using \: (1) \} \\ [/tex]

[tex]\rm \: 81 + 27d - 9d - {3d}^{2} = 105 \\ [/tex]

[tex]\rm \: 81 + 18d - {3d}^{2} = 105 \\ [/tex]

[tex]\rm \: {3d}^{2} - 18d + 105 - 81 = 0 \\ [/tex]

[tex]\rm \: {3d}^{2} - 18d + 24 = 0 \\ [/tex]

[tex]\rm \: 3({d}^{2} - 6d + 8) = 0 \\ [/tex]

[tex]\rm \: {d}^{2} - 6d + 8 = 0 \\ [/tex]

[tex]\rm \: {d}^{2} - 4d - 2d + 8 = 0 \\ [/tex]

[tex]\rm \: d(d - 4) - 2(d - 4) = 0 \\ [/tex]

[tex]\rm \: (d - 4)(d - 2) = 0 \\ [/tex]

[tex]\bf\implies \:d \: = \: 4 \: \: \: or \: \: \: d = 2 \\ [/tex]

So, two cases aries :-

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

[tex]\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} \\ [/tex]

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

[tex]\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} \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information :-

↝ nᵗʰ term of an arithmetic progression is,

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

↝ Sum of n  terms of an arithmetic progression is,

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

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.