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

Given that, the sum of three terms of an AP is 12. & The product of first and third term is 8 greater than second term.

So, Let's consider a - d , a , a + d be three terms of A.P.

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${\underline{\pmb{\frak{\bigstar\:According\:to\:the\:Question\::}}}}$

• Sum of first three terms of AP is 12.

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$\dashrightarrow\sf (a - d) + (a) + (a + d) = 12\\\\\\ \dashrightarrow\sf 3a\: \cancel{- \: d} \: \cancel{+\: d} = 12\\\\\\ \dashrightarrow\sf 3a = 12\\\\\\ \dashrightarrow\sf a = \cancel{\dfrac{12}{3}}\\\\\\ \dashrightarrow{\underline{\boxed{\pmb{\frak{a = 4}}}}}\:\bigstar$

$\therefore\:{\underline{\sf{Second\:term\:(a)\:of\:AP\:is\:{\pmb{4}}.}}}$

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▪︎Now, Let's solve second Condition —

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

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$\qquad\dashrightarrow\sf (a - d)(a + d) = a + 8\\\\\\ \qquad\dashrightarrow\sf (4 - d)(4 + d) = 4 + 8\\\\\\ \qquad\dashrightarrow\sf 4^2 - d^2 = 12\\\\\\ \qquad\dashrightarrow\sf 16 - d^2 = 12\\\\\\ \qquad\dashrightarrow\sf d^2 = 16 - 12\\\\\\ \qquad\dashrightarrow\sf d^2 = 4\\\\\\ \qquad\dashrightarrow{\underline{\boxed{\pmb{\frak{d = 2}}}}}\:\bigstar$

$\therefore\:{\underline{\sf{Common\:diffrence\:(d)\:between\:the\:terms\:of\:AP\:is\:{\pmb{2}}.}}}$

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❍ Therefore, The required terms of A.P. are :

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$\qquad\quad\twoheadrightarrow\sf (a - d) \:,\: a \:,\: (a + d)\\\\\\ \qquad\quad\twoheadrightarrow\sf (4 - 2) \:,\: 4 \:,\: (4 + 2)\\\\\\ \qquad\quad\twoheadrightarrow{\pmb{\frak{\purple{2 \:,\: 4 \:,\: 6}}}}$

Answer 2

Given :-

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

To Find :-

Terms

Solution :-

Let the terms of the AP be a - d, a, a + d

a - d + a + a + d = 12

(a + a + a) + (d - d) = 12

3a = 12

12/3 = a

4 = a

Now

(a - d) × (a + d) = a + 8

(4 - d) × (4 + d) = 8 + 4

(4 × 4) - (d × d) = 8 + 4

16 - d² = 8 + 4

-d² = 8 - 12

-d² = -4

d² = 4

d = √4

d = ±2

Finding the term

a - d = 4 - 2 = 2

a = 4

a + d = 4 + 2 = 6