Question

[Class 10, Arithmetic Progression]
Q: The sum of three numbers in AP is 18. If the product of first and third number is five times the common differnce, find the number.

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Answer 1

$\large\bf\underline{Given:-}$

• sum of three numbers in AP = 18
• product of 1st and 3rd number is 5 times the common difference.

$\large\bf\underline {To \: find:-}$

• we need to find the numbers.

$\huge\bf\underline{Solution:-}$

Let the numbers be (a - d), a (a + d)

Sum of three numbers = 18

⠀⠀⠀⠀⠀➝ (a-d) + a + (a+d) = 18

⠀⠀⠀⠀⠀➝ a -d + a + a + d = 18

⠀⠀⠀⠀⠀➝ 3a = 18

⠀⠀⠀⠀⠀➝ a = 18/3

⠀⠀⠀⠀⠀➝ a = 6

Product of 1st and 3rd numbers = 5d

⠀⠀⠀⠀⠀➝( a-d)(a +d) = 5d

By using identity :- (a+b)(a-b) = a² -b²

⠀⠀⠀⠀⠀➝ a² -d² = 5d

• a = 6

⠀⠀⠀⠀⠀➝ 6² -d² = 5d

⠀⠀⠀⠀⠀➝ 36 -d² = 5d

⠀⠀⠀⠀⠀➝ d² + 5d - 36 = 0

⠀⠀⠀⠀⠀➝ d² + 9d - 4d -36

⠀⠀⠀⠀⠀➝ d(d +9) -4( d+9)

⠀⠀⠀⠀⠀➝ (d -4)(d+9)

⠀⠀⠀⠀⠀➝ d = 4 or d = -9

• Let d = 4

the three numbers are :-

a - d = 6 -4 = 2

a = 4

a + d = 6 + 4 = 10

⠀⠀⠀⠀⠀

Three numbers are 2 , 4 , 10