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
Solution
<p>$\because$ a<sub>1</sub>, a<sub>2</sub>, .... a<sub>n</sub> be an A.P of natural numbers and</p>
<p>$${{{S_5}} \over {{S_9}}} = {5 \over {17}} \Rightarrow {{{5 \over 2}[2{a_1} + 4d]} \over {{9 \over 2}[2{a_1} + 8d]}} = {5 \over {17}}$$</p>
<p>$\Rightarrow 34{a_1} + 68d = 18{a_1} + 72d$</p>
<p>$\Rightarrow 16{a_1} = 4d$</p>
<p>$\therefore$ $d = 4{a_1}$</p>
<p>And $110 < {a_{15}} < 120$</p>
<p>$\therefore$ $110 < {a_1} + 14d < 120 \Rightarrow 110 < 57{a_1} < 120$</p>
<p>$\therefore$ ${a_1} = 2$ ($\because$ ${a_i}\, \in N$)</p>
<p>$d = 8$</p>
<p>$\therefore$ ${S_{10}} = 5[4 + 9 \times 8] = 380$</p>
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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