Question

Find three numbers in AP whose sum is 24 and the product is 480. Proper steps with correct answer will et brainliest

Answer 1

To Find:

• we need to find three numbers in AP . whose sum is 24 and product is 480.

Given :

• Product of Numbers = 480
• Sum of numbers = 24

Solution :

Let

Three numbers in AP be (a - d), a ,(a + d)

Now,

Sum of three numbers = 24

⟹ (a - d) + a + (a + d) = 24

⟹ a - d + a + a + d = 24

⟹ 3a = 24

⟹ a = 24/3

⟹ a = 8 ...1)

Now,

Product of three numbers = 480

⟹ (a + d) × (a - d) × (a) = 480

⟹ (a² - d²)a = 480

⟹ a³ - ad² = 480

• putting value of a from ...1)

⟹ 8³ - 8 × d² = 480

⟹ 512 - 8d² = 480

⟹ - 8d² = 480 - 512

⟹ - 8d² = -32

⟹ d² = 32/8

⟹ d = √4

⟹ d = ±2

So,

• Three numbers in AP are :-

When d is positive (+)

⇛(a - d) = 8 - 2 = 6

⇛a = 8

⇛a + d = 8 + 2 = 10

When d is negative (-)

⇛ (a - d) = 8 -(-2) = 10

⇛ a = 8

⇛ (a + d) = 8 - 2 = 6

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Answer 2

Let the three terms of the AP be

a-d, a and a+d, so that

a-d+ a + a+d = 24, or

a = 24/3 = 8 …(1)

Their product = (a-d)(a)(a+d) = 480, or

a(a^2-d^2)= 480, or

(a^2-d^2) = 480/8 = 60

8^2-d^2=60, or

64–60 = 4 = d^2, or d = +2 or -2.

Hence the three terms of the AP are 6, 8 and 10, or 10, 8 and 6.

Hence the three terms of the AP are 6, 8 and 10, or 10, 8 and 6.

f a is the first and d the difference we have the sequence: a, $a+d, a+2d$

Sum =  thus = middle value

Prime decomposition of $480=2^5x3x5$

Product of $a(a+d)(a+2d)=480$ and$a+d=8$

implies $a(a+2d)=2^2x3x5$

Enumerating the possibilities of two numbers a, a+2d centred on 8 whose product is $2^2x3x5$ we conclude $a=6, a+2d=10$

Finally this gives the arithmetic sequence: 6,8,10

hope it helps

:)