Question

(i) In an A.P., the sum of first n terms is $\frac{3n^2}{2}+\frac{13}{n}n.$ Find its 25^th term.

(ii) If the sum of first n terms of an A.P. is n², then find its 10th term.

Answer 1

(i)

Here, the sum of first n terms is given by the expression,

We need to find the 25^th term of the A.P.

So we know that the n^thterm of an A.P. is given by,

So …… (1)

So, using the expression given for the sum of n terms, we find the sum of 25 terms (S₂₅) and the sum of 24 terms (S₂₄). We get,

Similarly,

Now, using the above values in (1),

Therefore, .

(ii) Given:
$$\begin{gathered} S_n=n^2 \\ \mathrm{Now}, \\ \mathrm{Sum}\mathrm{of}\mathrm{one}\mathrm{term}=S_1=1^2 \\ \Rightarrow a_1=a=1...\left(1\right) \\ \mathrm{Sum}\mathrm{of}\mathrm{two}\mathrm{term}s=S_2=2^2 \\ \Rightarrow a_1+a_2=4...\left(2\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get} \\ {a}_2=3 \\ \Rightarrow a+d=3 \\ \Rightarrow d=2\left(\because a=1\right) \\ \mathrm{Therefore}, \\ {a}_{10}=a+\left(10-1\right)d \\ =1+9\left(2\right) \\ =19 \end{gathered}$$

Hence, the 10th term is 19.