Question

Finf x',if 3x2+4x+4,2x2+3x+3 and 3x+8 are term of ap where x is a natural number

Answer 1

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

Let assume that

$\rm :\longmapsto\:\: a_1 = {3x}^{2} + 4x + 4$

$\rm :\longmapsto\:\: a_2 = {2x}^{2} + 3x + 3$

$\rm :\longmapsto\:\: a_3 = 3x + 8$

Since, it is given that

$\rm :\longmapsto\:a_1, \: a_2, \: a_3 \: are \: in \: AP$

$\rm :\implies\:a_2 - a_1 = a_3 - a_2$

$\rm :\longmapsto\: {2x}^{2} + 3x + 3 - {3x}^{2} - 4x - 4 = 3x + 8 - {2x}^{2} - 3x - 3$

$\rm :\longmapsto\: - {x}^{2} - x - 1 = - {2x}^{2} + 5$

$\rm :\longmapsto\: {x}^{2} - x - 6 = 0$

$\rm :\longmapsto\: {x}^{2} - 3x + 2x - 6 = 0$

$\rm :\longmapsto\:x(x - 3) + 2(x - 3) = 0$

$\rm :\longmapsto\:(x - 3)(x + 2) = 0$

$\bf\implies \:x = 3 \: or \: x = - 2 \: (rejected \: as \: x \in \: natural \: number)$

$\bf\implies \:x = 3$

Additional Information :-

↝ nᵗʰ term of an arithmetic sequence is,

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

↝ Sum of n term of an arithmetic sequence is,

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

Or

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

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.