Question

find three numbers in Ap whose sum is 15 and product is 80​

Answer 1

Answer:

2,5,8 or 8,5,2

Step-by-step explanation:

let a -d, a, a+d are three terms of an ap according to the problem

Answer 2

Given

• Three numbers are in A.P
• Their sum is 15 and product is 80.

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To find

• The three numbers.

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Solution

• The three terms are written as :-

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→ $\sf{a_1 = a}$

→ $\sf{a_2 = a + d}$

→ $\sf{a_3 = a + 2d}$

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⠀⠀❍ According to the question

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★ Sum of these three terms is 15.

$\tt\longmapsto{a + (a + d) + (a + 2d) = 15}$

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$\tt\longmapsto{3a + 3d = 15}$

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$\tt\longmapsto{a + d = 5}$⠀⠀

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$\tt\longmapsto{a = 5 - d}$⠀⠀.....[1]

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★ Product of these three terms is 80.

$\tt\longmapsto{a \times (a + d) \times (a + 2d) = 80}$

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⠀⠀❍ Putting value of a from [1]

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$\tt\longmapsto{(5 - d) \times (5 - \cancel{d} + \cancel{d}) \times (5 - d + 2d) = 80}$

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$\tt\longmapsto{(5 - d) \times 5 \times (5 + d) = 80}$

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$\tt\longmapsto{(5 - d) \times (5 + d) = \dfrac{80}{5}}$

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$\tt\longmapsto{(5 - d) (5 + d) = 16}$

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⠀❍ Using identity

$\: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{\bigstar{a^2 - b^2 = (a + b) (a - b){\bigstar}}}}$

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$\tt\longmapsto{(5)^2 - (d)^2 = 16}$

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$\tt\longmapsto{25 - d^2 = 16}$

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$\tt\longmapsto{- d^2 = 16 - 25}$

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$\tt\longmapsto{\cancel{-} d^2 = \cancel{-} 9}$

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$\tt\longmapsto{d = \sqrt{9}}$

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$\tt\longmapsto{d = ± 3}$

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⠀⠀❍ Putting the value of d in eq[1]

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$\boxed{\sf{\orange{Case\: 1 \rightarrow d = 3}}}$

$\tt:\implies{a = 5 - 3}$

$\tt:\implies{a = 2}$

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$\boxed{\sf{\orange{Case\: 2 \rightarrow d = -3}}}$

$\tt:\implies{a = 5 + 3}$

$\tt:\implies{a = 8}$

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⠀⠀❍ We have

• a = 2 or 8
• d = 3 or -3

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Hence,

$\bf:\implies\: \: \: \: \: \: \: \: {Required\: A.P.}$

• A.P → 2, 5, 8

OR

• A.P → 8, 5, 2

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