Question

The sum of three terms of an AP is 12 The product of first and third term is 8 greater than second term find the terms
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Answer 1

Answer:

Step-by-step explanation:

Answer 2

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

★ Let us first assume three terms of AP series.

$\begin{gathered}\begin{gathered}\bf\: Let-\begin{cases} &\sf{first \: term = a - d} \\ &\sf{second \: term = a}\\ &\sf{third \: term \: = a + d} \end{cases}\end{gathered}\end{gathered}$

According to first condition,

★ Sum of three terms of an AP = 12

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

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

$\bf\implies \:a = 4 - - (1)$

According to second condition,

★ The product of first and third term is 8 greater than second term.

$\rm :\longmapsto\:(a - d)(a + d) - a = 8$

$\rm :\longmapsto\:(4 - d)(4 + d) - 4 = 8$

$\rm :\longmapsto\:16 - {d}^{2} = 8 + 4$

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

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

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

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

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

Hence, two cases arises

★ Case :- 1 When a = 4 and d = 2

$\begin{gathered}\begin{gathered}\bf\: Hence-\begin{cases} &\sf{first \: term = a - d = 4 - 2 = 2} \\ &\sf{second \: term = a = 4}\\ &\sf{third \: term \: = a + d = 4 + 2 = 6} \end{cases}\end{gathered}\end{gathered}$

★ Case :- 2 When a = 4 and d = - 2

$\begin{gathered}\begin{gathered}\bf\: Hence-\begin{cases} &\sf{first \: term = a - d = 4 + 2 = 6} \\ &\sf{second \: term = a = 4}\\ &\sf{third \: term \: = a + d = 4 - 2 = 2} \end{cases}\end{gathered}\end{gathered}$

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.