Question

The value of x for which 2x, (x + 10) and (3x + 2) are the three consecutive terms of an AP, is
(a) 6 (b) -6 (c) 18 (d) -18.

With proper explanation.​

Answer 1

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

Three consecutive terms of an AP are 2x, x + 10 and 3x + 2

$\large\underline{\sf{To\:Find - }}$

The value of x.

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

Given three consecutive terms of an AP is 2x, x + 10 and 3x + 2.

Let assume that

$\rm :\longmapsto\:a_1 = 2x$

$\rm :\longmapsto\:a_2 = x + 10$

$\rm :\longmapsto\:a_3 = 3x + 2$

We know that,

$\rm :\longmapsto\:a_1,a_2,a_3 \: forms \: arithmetic \: progression \: if$

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

$\rm :\longmapsto\:x + 10 - 2x = 3x + 2 - (x + 10)$

$\rm :\longmapsto\:10 - x = 3x + 2 - x - 10$

$\rm :\longmapsto\:10 - x = 2x - 8$

$\rm :\longmapsto\: - x - 2x = - 10 - 8$

$\rm :\longmapsto\: - 3x = - 18$

$\bf :\longmapsto\: x = 6$

Verification,

$\rm :\longmapsto\:a_1 = 2x = 2 \times 6 = 12$

$\rm :\longmapsto\:a_2 = x + 10 = 6 + 10 = 16$

$\rm :\longmapsto\:a_3 = 3 \times 6 + 2 = 20$

Now,

$\rm :\longmapsto\:a_2 - a_1 = 16 - 12 = 4$

$\rm :\longmapsto\:a_3 - a_2 = 20 - 16 = 4$

Additional Information :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

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

and

↝ 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.