Question

10. If the sum of three consecutive terms of an A.P.be 12, the product of 1st and the 3rd is 3 times of the 2nd, find the three terms.​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

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

Let assume that

★ Three consecutive terms of AP as

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{a - d} \\ \\ &\sf{a} \\ \\ &\sf{a + d} \end{cases}\end{gathered}\end{gathered}$

According to statement,

The sum of three consecutive terms of an A.P.be 12

$\rm :\longmapsto\:a - d + a + a + d = 12$

$\rm :\longmapsto\:3a = 12$

$\bf\implies \:a = 4$

According to statement again,

The product of 1st and the 3rd is 3 times of the 2nd term.

$\rm :\longmapsto\:(a - d)(a + d) = 3a$

On substituting the value of a, we get

$\rm :\longmapsto\:(4 - d)(4 + d) = 12$

$\rm :\longmapsto\:16 - {d}^{2} = 12$

$\rm :\longmapsto\: - {d}^{2} = 12 - 16$

$\rm :\longmapsto\: - {d}^{2} = - 4$

$\rm :\longmapsto\: {d}^{2} = 4$

$\bf\implies \:d \: = \: \pm \: 2$

So, 2 cases arises.

Case : 1

When a = 4 and d = 2, we get the three terms as

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{a - d = 4 - 2 = 2} \\ \\ &\sf{a = 4} \\ \\ &\sf{a + d = 4 + 2 = 6} \end{cases}\end{gathered}\end{gathered}$

Case : 2

When a = 4 and d = - 2, we get three terms as

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{a - d = 4 + 2 = 6} \\ \\ &\sf{a = 4} \\ \\ &\sf{a + d = 4 - 2 = 2} \end{cases}\end{gathered}\end{gathered}$

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Additional Information

↝ nᵗʰ term of an arithmetic sequence is,

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

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.