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8. Sequences and Series
hard
Suppose $a_{1}, a_{2}, \ldots, a_{ n }, \ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms of the sum of first nine terms of the progression is $5: 17$ and $110< a_{15} < 120$ , then the sum of the first ten terms of the progression is equal to -
A
$290$
B
$380$
C
$460$
D
$510$
(JEE MAIN-2022)
Solution
$\frac{ S _{5}}{ S _{9}}=\frac{5}{17} \Rightarrow \frac{\frac{5}{2}(2 a+4 d)}{\frac{9}{2}(2 a+8 d)}=\frac{5}{17}$
$\Rightarrow d=4\,a$
$a_{15}=a+14 d=57\,a$
Now, $110< a _{15}<120$
$110<57\,a < 120$
$a =2 \therefore d =8$
$S _{10}=\frac{10}{2}(2 \times 2+9 \times 8)=380$
Standard 11
Mathematics