The number of terms in an A.P.

Question

The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that of the even terms is 30. If the last term exceeds the first term by $10	frac12$, then the number of terms in the A.P. is:

Accepted Answer

Let there be $2n$ terms with first term $a$, common difference $d$. Odd-position terms ($n$ of them) sum to
$$n[a+(n-1)d]=24 \ \ (i)$$
Even-position terms sum to
$$n[a+d+(n-1)d]=30 \ \ (ii)$$

Subtracting: $nd=6$ ... (iii)

Last term exceeds first by $\dfrac{21}{2}$: $a+(2n-1)d-a=\dfrac{21}{2}\Rightarrow 2nd-d=\dfrac{21}{2}\Rightarrow 12-d=\dfrac{21}{2}\Rightarrow d=\dfrac32.$

From (iii): $n=\dfrac{6}{3/2}=4$. Total terms $=2n=8$.

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