If a1 (> 0), a2, a3, a4, a5 are in a G.P., a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4, then a2 + a4 + 2a5 is equal to ___________.
Answer (integer)
40
Solution
<p>Let G.P. be a<sub>1</sub> = a, a<sub>2</sub> = ar, a<sub>3</sub> = ar<sup>2</sup>, .........</p>
<p>$\because$ 3a<sub>2</sub> + a<sub>3</sub> = 2a<sub>4</sub></p>
<p>$\Rightarrow$ 3ar + ar<sup>2</sup> = 2ar<sup>3</sup></p>
<p>$\Rightarrow$ 2ar<sup>2</sup> $-$ r $-$ 3 = 0</p>
<p>$\therefore$ r = $-$1 or ${3 \over 2}$</p>
<p>$\because$ a<sub>1</sub> = a > 0 then r $\ne$ $-$1</p>
<p>Now, a<sub>2</sub> + a<sub>4</sub> = 2a<sub>3</sub> + 1</p>
<p>ar + ar<sup>3</sup> = 2ar<sup>2</sup> + 1</p>
<p>$a\left( {{3 \over 2} + {{27} \over 8} - {9 \over 2}} \right) = 1$</p>
<p>$\therefore$ a = ${8 \over 3}$</p>
<p>$\therefore$ a<sub>2</sub> + a<sub>4</sub> + 2a<sub>5</sub> = a(r + r<sup>3</sup> + 2r<sup>4</sup>)</p>
<p>$= {8 \over 3}\left( {{3 \over 2} + {{27} \over 8} + {{81} \over 8}} \right) = 40$</p>
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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