How many terms of the sequence 18, 16, 14…. should be taken so tha

1.	How many terms of the sequence 18, 16, 14…. should be taken so that their sum is zero.
Answer» Given AP. is 18, 16, 14, … We know that, S_n = \(\frac{n}{2}\)[2a + (n − 1)d] Here, The first term (a) = 18 The sum of n terms (S_n) = 0 (given) Common difference of the A.P. (d) = a₂ – a₁ = 16 – 18 = – 2 So, on substituting the values in S_n ⟹ 0 = \(\frac{n}{2}\)[2(18) + (n − 1)(−2)] ⟹ 0 = \(\frac{n}{2}\)[36 + (−2n + 2)] ⟹ 0 = \(\frac{n}{2}\)[38 − 2n] Further, \(\frac{n}{2}\) ⟹ n = 0 Or, 38 – 2n = 0 ⟹ 2n = 38 ⟹ n = 19 Since, the number of terms cannot be zero, hence the number of terms (n) should be 19.

How many terms of the sequence 18, 16, 14…. should be taken so that their sum is zero.

Answer»

Given AP. is 18, 16, 14, …

We know that,

S_n = \(\frac{n}{2}\)[2a + (n − 1)d]

Here,

The first term (a) = 18

The sum of n terms (S_n) = 0 (given)

Common difference of the A.P.

(d) = a₂ – a₁ = 16 – 18 = – 2

So, on substituting the values in S_n

⟹ 0 = \(\frac{n}{2}\)[2(18) + (n − 1)(−2)]

⟹ 0 = \(\frac{n}{2}\)[36 + (−2n + 2)]

⟹ 0 = \(\frac{n}{2}\)[38 − 2n] Further, \(\frac{n}{2}\)

⟹ n = 0 Or, 38 – 2n = 0

⟹ 2n = 38

⟹ n = 19

Since, the number of terms cannot be zero, hence the number of terms (n) should be 19.