JEE MAIN - Mathematics (2022 - 27th July Morning Shift - No. 6)
Suppose $$a_{1}, a_{2}, \ldots, a_{n}$$, .. be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to 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
290
380
460
510
Explanation
$$\because$$ a1, a2, .... an be an A.P of natural numbers and
$${{{S_5}} \over {{S_9}}} = {5 \over {17}} \Rightarrow {{{5 \over 2}[2{a_1} + 4d]} \over {{9 \over 2}[2{a_1} + 8d]}} = {5 \over {17}}$$
$$ \Rightarrow 34{a_1} + 68d = 18{a_1} + 72d$$
$$ \Rightarrow 16{a_1} = 4d$$
$$\therefore$$ $$d = 4{a_1}$$
And $$110 < {a_{15}} < 120$$
$$\therefore$$ $$110 < {a_1} + 14d < 120 \Rightarrow 110 < 57{a_1} < 120$$
$$\therefore$$ $${a_1} = 2$$ ($$\because$$ $${a_i}\, \in N$$)
$$d = 8$$
$$\therefore$$ $${S_{10}} = 5[4 + 9 \times 8] = 380$$
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