Question

If the sum of n terms of an ap is represented by6n-n^2 ,the 10th term of ap

Answer 1

Question: If the sum of 'n' terms of an AP is represented by 6n - n², find the 10th term of an AP.

Before answering this question, make sure you've understood this.

The First term of an AP is equal to S₁

The Second term of an AP is equal to S₂ - S₁

The Third term of an AP is equal to S₃ - S₂

And so on.

Solution.

Sum of 'n' terms represents $\sf S_n$ of the given AP.

Here, $\sf S_n$ is represented by 6n - n²

$\sf S_n = 6n - n^{2}$

Let n = 1

$\sf S_1 = 6(1) - (1)^{2}$

$\sf S_1 = 6 - 1$

$\sf S_1 = 5$ .....Eq(1)

$\sf S_n = 6n - n^{2}$

Let n = 2

$\sf S_2 = 6(2) - (2)^{2}$

$\sf S_2 = 12 - 4$

$\sf S_2 = 8$ .....Eq(2)

$\sf S_n = 6n - n^{2}$

Let n = 3

$\sf S_3 = 6(3) - (3)^{2}$

$\sf S_3 = 18 - 9$

$\sf S_3 = 9$ .....Eq(3)

We know that,

a₁ = S₁

a₁ = 5 [From Eq 1]

We also know that,

a₂ = S₂ - S₁

a₂ = 8 - 5 [From Eq1 and Eq2]

a₂ = 3

a₃ = S₃ - S₂

a₃ = 9 - 8

a₃ = 1

Now, we find the common difference,

a₂ - a₁ = a₃ - a₁

3 - 5 = 1 - 3

-2 = -2

d = -2

Now, We find the 10th term.

$\sf a_n = a + (n-1)d$

a = 5, d = -2, n = 10

$\sf a_{10} = 5 + (10-1)-2$

$\sf a_{10} = 5 + (9)-2$

$\sf a_{10} = 5 - 18$

$\sf a_{10} = -13$

Therefore, the 10th term of the given AP is -13.