Question

How many terms of the arithmetic sequence 5, 7, 9, …. should be added to get 140?

Answer 1

Let the terms of the sequence be x₁, x₂, x₃, …

We know that the terms of an A.P. are of the form a + b, 2a + b, 3a + b, …

The given sequence is 5, 7, 9, …

∴ First term, x₁ = a + b = 5

Second term, x₂ = 2a + b = 7

⇒ a + a + b = 7

⇒ a + 5 = 7

⇒ a = 7 − 5 = 2

b = 5 − a = 5 − 2 = 3

Let the sum of n terms of the given sequence be 140.

Let the n^th term of the sequence be x_n.

⇒ x_n = an + b = 2n + 3

S_n = 140

⇒

⇒

Adding 4 to both the sides to make L.H.S a perfect square:

⇒ + 4 = 140 + 4

⇒

⇒ {Using a² + b² + 2ab = (a + b)²}

Since n is a natural number, it cannot be negative.

⇒ n = 10

Thus, 12 terms of the given sequence should be added to get 140.