Sequences and Series 2 Question 6
7. The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24 , then what is the length of its smallest side?
(2017 Adv.)
(a) -153
(b) -133
(c) -131
(d) -135
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Answer:
Correct Answer: 7. (c)
Solution:
- We have, $S=a _1+a _2+\ldots+a _{30}$
$$ =15\left[2 a _1+29 d\right] $$
(where $d$ is the common difference)
$$ \because S _n=\frac{n}{2}[2 a+(n-1) d] $$
and
$$ \begin{aligned} T & =a _1+a _3+\ldots+a _{29} \\ & \left.=\frac{15}{2}\left[2 a _1+14 \times 2 d\right)\right] \end{aligned} $$
( $\because$ common difference is $2 d$ )
$$ \Rightarrow \quad 2 T=15\left[2 a _1+28 d\right] $$
From Eqs. (i) and (ii), we get
$$ S-2 T=15 d=75 \quad[\because S-2 T=75] $$
$$ \Rightarrow \quad d=5 $$
Now, $\quad a _{10}=a _5+5 d$
$$ =27+25=52 $$
