How many terms of the A.P. is 27, 24, 21. . . should be taken that the

1.	How many terms of the A.P. is 27, 24, 21. . . should be taken that their sum is zero?
Answer» Given A.P. is 27, 24, 21. . . We know that, S_n = \(\frac{n}{2}\)[2a + (n − 1)d] Here we have, the first term (a) = 27 The sum of n terms (S_n) = 0 Common difference of the A.P. (d) = a₂ – a₁ = 24 – 27 = -3 On substituting the values in S_n, we get ⟹ 0 = \(\frac{n}{2}\)[2(27) + (n − 1)( − 3)] ⟹ 0 = (n)[54 + (n – 1)(-3)] ⟹ 0 = (n)[54 – 3n + 3] ⟹ 0 = n [57 – 3n] Further we have, n = 0 Or, 57 – 3n = 0 ⟹ 3n = 57 ⟹ n = 19 The number of terms cannot be zero, Hence, the numbers of terms (n) is 19.

How many terms of the A.P. is 27, 24, 21. . . should be taken that their sum is zero?

Answer»

Given A.P. is 27, 24, 21. . .

We know that,

S_n = \(\frac{n}{2}\)[2a + (n − 1)d]

Here we have, the first term (a) = 27

The sum of n terms (S_n) = 0

Common difference of the A.P. (d) = a₂ – a₁ = 24 – 27 = -3

On substituting the values in S_n, we get

⟹ 0 = \(\frac{n}{2}\)[2(27) + (n − 1)( − 3)]

⟹ 0 = (n)[54 + (n – 1)(-3)]

⟹ 0 = (n)[54 – 3n + 3]

⟹ 0 = n [57 – 3n] Further we have, n = 0 Or, 57 – 3n = 0

⟹ 3n = 57

⟹ n = 19

The number of terms cannot be zero,

Hence, the numbers of terms (n) is 19.