GMAT Practice Question: The sequence a_{1}, a_{2}, \ldots a_{n}, \ldots is such that a_{n}=2 a_{n-1}-x...
Question
The sequence a_{1}, a_{2}, \ldots a_{n}, \ldots is such that a_{n}=2 a_{n-1}-x for all positive integers n \geq 2 and for a certain number x. If a_{5}=99 and a_{3}=27, what is the value of x ?
- 3
- 9
- 18
- 36
- 45
Topics: linear equations, sequences & series
Solution
Step 1
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 \times 27 - x a_4 = 54 - x a_5 = 2a_4 - x a_5 = 2 \times (54 - x) - x a_5 = 108 - 3x
Step 2
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 = \frac{-9}{-3} x = 3
Answer
3
