Question

The sum of 5 numbers in AP is 10 and their product is 80. What are the five numbers?

(a) 2 + 2√6, 2 + √6, 2, 2 – 2√6, 2 – √6

(b) 2 + 2√6, 2 + √6, 2 – 2√6, 2 – √6

(c) 2 + 2√6, 2 – √6, 2, 2 – 2√6, 2 – √6

(d) 2 + 2√6, 2, 2 – 2√6, 2 – √6

This question was addressed to me in semester exam.

My question is from Arithmetic Progression Basics topic in section Arithmetic Progressions of Mathematics – Class 10

Answer 1

Right choice is (a) 2 + 2√6, 2 + √6, 2, 2 – 2√6, 2 – √6

The explanation is: Let the five numbers in AP be (a + 2d), (a + d), a, (a – 2d) and (a – d).

Sum of the five numbers is 10.

∴ (a + 2d) + (a + d) + a + (a – 2d) + (a – d) = 10

5a = 10

a = 2

Now, product of the numbers is 80.

∴ (a + 2d)(a + d)a(a – 2d)(a – d) = 80

a^5 – 5a^3 d^2 + 4d^4 a = 80

Substituting a = 2, we get

2^5 – 5(2^3 d^2) + 4d^4 (2) = 80

32 – 40d^2 + 8d^4 = 80

8d^4 – 40d^2 – 48 = 0

Substitute d^2 = t

t^2 – 5t – 6 = 0

Solving the equation, we get,

t = 6, -1

Now, t = d^2

Hence, d = ±√6, ±√-1

The five numbers are 2 + 2√6, 2 + √6, 2, 2 – 2√6, 2 – √6 or 2 – 2√6, 2 – √6, 2, 2 + 2√6, 2 + √6 or 2 + 2√-1, 2 + √-1, 2, 2 – 2√-1, 2 – √-1 or 2 – 2√-1, 2 – √-1, 2, 2 + 2√-1, 2 + √-1.