Consider an A. P. of positive integers, whose sum of the first...

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its (11^{\text{th }}) term is :

Options

A90
B84
C122
D108

Correct answer

A. 90

Step-by-step solution

\begin{aligned} & \mathrm{S}_3=3 \mathrm{a}+3 \mathrm{~d}=54 \\ & \Rightarrow \mathrm{a}+\mathrm{d}=18 \\ & \mathrm{~S}_{20}=10(2 \mathrm{a}+19 \mathrm{~d}) \\ & \Rightarrow 10(36+17 \mathrm{~d}) \\ & \Rightarrow 1600 < 10(36+17 \mathrm{~d}) < 1800 \\ & \Rightarrow 160 < 36+17 \mathrm{~d} < 180 \\ & \Rightarrow 124 < 17 \mathrm{~d} < 144 \\ & \Rightarrow 7 \frac{5}{17} < \mathrm{d} < 8 \frac{8}{17} \end{aligned}

Common difference will be natural number

\begin{aligned} & \Rightarrow d=8 \Rightarrow a=10 \\ & \Rightarrow a_{11}=10+10 \times 8=90 \end{aligned}

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