Question

if x+2,3x+2 and 7x-2 are three terms of an AP . write its algebric expression​

Answer 1

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

Given that,

$\rm \: x + 2, \: 3x + 2, \: 7x - 2 \: are \: in \: AP - - - - (1)$

We know, three numbers a, b, c are in AP iff

$\boxed{\sf{  \:\rm \: b - a \: = \: c - b \: \: }}$

So, using this result, we get

$\rm \: 3x + 2 - (x + 2) = 7x - 2 - (3x + 2)$

$\rm \: 3x + 2 - x - 2 = 7x - 2 - 3x - 2$

$\rm \: 2x = 4x - 4$

$\rm \: 2x - 4x = - 4$

$\rm \: - 2x = - 4$

$\rm\implies \:x = 2$

Thus, on substituting the value of x in equation (1), we have

$\rm \: 2 + 2, \: 3(2) + 2, \: 7(2) - 2 \: are \: in \: AP$

$\rm \: 4, \: 6 + 2, \: 14 - 2 \: are \: in \: AP$

$\rm \: 4, \: 8, \: 12 \: are \: in \: AP$

So, it means,

First term of an AP series, a = 4

Common difference of an AP series, d = 8 - 4 = 4

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ general term or nᵗʰ term of an arithmetic progression 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.

Tʜᴜs,

↝ general term or nᵗʰ term is,

$\rm \: a_n = a + (n - 1)d$

On substituting the values of a and d, we get

$\rm \: a_n = 4 + (n - 1)4$

$\rm \: a_n = 4 + 4n - 4$

$\rm\implies \:a_n \: = \: 4n \:$

Hence,

The general algebraic expression is

$\rm\implies \: \boxed{\sf{  \:\rm \: \:a_n \: = \: 4n \: \: \: }}$

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Additional Information :-

↝ 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.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.