Question

How many terms of the A.P 1, 4, 7,... are needed to give the sum 925?

• A. 20
• B. 22
• C. 24
• D. 25

Answer 1

The correct option is D 25

Let n = 24, then
$T_{24}=1+23\times 3=70\therefore S_{24}=\left(\frac{1+70}{2}\right)\times 24=71\times 12=852\therefore S_{24}=852<925,$
Thus it is obvious that n must be greater than 24, which gives us n = 25, as per the choices given.
$\therefore S_{25}$ = $\left(\frac{a+T_{25}}{2}\right)\times 25$ $because{T}_{25}=T_{24}+d=73$
= $\frac{1+73}{2}\times 25=925$
Hence (d) is correct.

Alternatively: $S_n=925=\frac{n}{2}[2\times 1+(n-1\left)3\right]\Rightarrow 1850=2n+3n^2-3n\Rightarrow 3n^2-n-1850=0\Rightarrow 3n^2-75n+74n-1850=0\Rightarrow 3n(n-25)+74(n-25)=0\Rightarrow n=25orn=-\frac{74}{3}$
The only admissible value of n = 25
Since number of terms cannot be negative and as a fraction too.
Hence (d).