Question

AP given that the first term (a) = 54, the common difference
(d) = -3 and the nth term (an) = 0, find n and the sum of first n terms (Sn)
of the A.P.​

Answer 1

$\textbf{Formula used:}$

$\text{The n th term of A.P is}$

$\boxed{\bf\,t_n=a+(n-1)d}$

$\text{The sum n terms of A.P is}$

$\boxed{\bf\,S_n=\frac{n}{2}[a+l]}$

$\textbf{Given:}$

$\text{First term, a=54}$

$\text{Common difference, d=-3 and}$

$t_n=0$

$\implies\,a+(n-1)d=0$

$\implies\,54+(n-1)(-3)=0$

$\implies\,3(n-1)=54$

$\implies\,n-1=18$

$\implies\boxed{\bf\,n=19}$

$\text{Now,}$

$\textbf{The sum of 19 terms}$

$=S_{19}$

$=\displaystyle\frac{n}{2}[a+l]$

$=\displaystyle\frac{19}{2}[54+0]$

$=19{\times}27$

$=513$

$\therefore\textbf{The value of n is 19 and}$

$\textbf{the sum of 19 terms is 513}$

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