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JEE Advanced 2014Paper 1
Exam
JEE Advanced
Year
2014
Paper
Paper 1
Subject
Mathematics
Chapter
Sequences and Series

JEE Advanced 2014 · Paper 1 · Mathematics · Sequences and Series

Let a, b, c be positive integers such that (b)/(a) is an integer....

Let a, b, c be positive integers such that ba\dfrac{b}{a} is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of a2+a−14a+1\dfrac{{a}^{2} + a - 14}{a + 1} is ________

Correct answer

0

Step-by-step solution

\dfrac{b}{c} = \dfrac{c}{b} = (integer)\left(integer\right)
b2=ac⇒c=b2a{b}^{2} = ac ⇒ c = \dfrac{{b}^{2}}{a}
a+b+c3=b+2\dfrac{a + b + c}{3} = b + 2
a+b+c=3b+6⇒a−2b+c=6a + b + c = 3b + 6 ⇒ a - 2b + c = 6
a−2b+b2a=6⇒1−2ba+b2a2=6aa - 2b + \dfrac{{b}^{2}}{a} = 6 ⇒ 1 - \dfrac{2b}{a} + \dfrac{{b}^{2}}{{a}^{2}} = \dfrac{6}{a}
(ba−1)2=6a⇒a=6{\left(\dfrac{b}{a} - 1\right)}^{2} = \dfrac{6}{a} ⇒ a = 6 only
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