The sequence a_1, a_2, a_n, is such that a_n=2 a_n-1-x...

Question

The sequence a_1, a_2, a_n, is such that a_n=2 a_n-1-x for all positive integers n 2 and for a certain number x. If a_5=99 and a_3=27, what is the value of x ?

GMAT Panda · Accepted Answer

3 We use the recurrence a_n = 2a_n-1 - x and the given values a_3 = 27 and a_5 = 99 to express the fifth term in terms of the third term and x by stepping forward from index 3 to index 5. a_4 = 2a_3 - x a_4 = 2 27 - x a_4 = 54 - x a_5 = 2a_4 - x a_5 = 2 (54 - x) - x a_5 = 108 - 3x We set this expression for the fifth term equal to the given value a_5 = 99 and solve for x. 108 - 3x = 99 -3x = 99 - 108 -3x = -9 x = -9/-3 x = 3

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$We use the recurrence a_{n} = 2 a_{n - 1} - x and the given values a_{3} = 27 and a_{5} = 99 to express the fifth term in terms of the third term and x by stepping forward from index 3 to index 5.$	$a_{4} = 2 a_{3} - x a_{4} = 2 \times 27 - x a_{4} = 54 - x a_{5} = 2 a_{4} - x a_{5} = 2 \times (54 - x) - x a_{5} = 108 - 3 x$
$We set this expression for the fifth term equal to the given value a_{5} = 99 and solve for x.$	$108 - 3 x = 99 - 3 x = 99 - 108 - 3 x = - 9 x = \frac{- 9}{- 3} x = 3$

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