Question

If 7x -2 ,3x-2 ,x+2 are consecutive terms of an AS ,then
a) Find the terms
b) Find the common difference

Answer 1

Appropriate Question

If 7x -2 ,3x-2 ,x+2 are consecutive terms of an AP, then

a) Find the terms

b) Find the common difference

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

Given that,

$\red{\rm :\longmapsto\:7x -2, \: 3x-2, \: x+2 are \: in \: AP}$

We know,

Three terms a, b, c are in AP iff b - a = c - b, i.e. common difference between the consecutive terms is same.

So, using this, we get

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

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

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

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

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

$\rm \implies\:\boxed{\tt{ \: x \: = \: - \: 2 \: }}$

So, terms of AP are as follow

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{7x - 2 = - 14 - 2 = - 16} \\ \\ &\sf{3x - 2 = - 6 - 2 = - 8} \\ \\ &\sf{x + 2 = - 2 + 2 = 0} \end{cases}\end{gathered}\end{gathered}$

So,

Common difference, d = - 8 - (- 16) = - 8 + 16 = 8

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↝ 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

Appropriate Question

If 7x -2 ,3x-2 ,x+2 are consecutive terms of an AP, then

a) Find the terms

b) Find the common difference

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

Given that,

$\red{\rm :\longmapsto\:7x -2, \: 3x-2, \: x+2 are \: in \: AP}$

We know,

Three terms a, b, c are in AP iff b - a = c - b, i.e. common difference between the consecutive terms is same.

So, using this, we get

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

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

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

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

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

$\rm \implies\:\boxed{\tt{ \: x \: = \: - \: 2 \: }}$

So, terms of AP are as follow

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{7x - 2 = - 14 - 2 = - 16} \\ \\ &\sf{3x - 2 = - 6 - 2 = - 8} \\ \\ &\sf{x + 2 = - 2 + 2 = 0} \end{cases}\end{gathered}\end{gathered}$

So,

Common difference, d = - 8 - (- 16) = - 8 + 16 = 8

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Explore more

↝ 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.