reto wrote:

In a certain series, each term (except for the first term) is one greater than twice the previous term. If the fifth term is equal to the third term, then the first term is which of the following?

A. −2

B. −1

C. 0

D. 1

E. 2

Working from answer choices is fast.

For n > 1, \(A_{n} = 2(A_{n-1}) + 1\)

\(A_1 = ??\)

\(A_2 = 2(A_1) + 1\)

\(A_3 = 2(A_2) + 1\) . .

I didn't write all that out; just the rule.

There are a couple of ways to conclude that the first term can't be 0, 1, or 2.

One way is testing. Start with (C) where

\(A_1 = 0\)

\(A_2 = 1\)

\(A_3 = 3\). Stop.

The sequence's values are increasing. Nothing in the rule will reverse that. Doubling and adding 1 means more successive increase. If 0 as a first term leads to successive increase, first terms of 1 and 2 make the growth worse.

OR: Logically, eliminate C, D, and E. This rule's operation includes doubling and adding +1. Both operations = successive increase of positive integers.

\(A_5\) will be greater than, not equal to, \(A_3\).

If not sure about this growth pattern, test just 0, as above.

Answers A and B are left. Try (B) -1:

\(A_1 = -1\)

\(A_2 = -1\)

\(A_3 = -1\)

There is the pattern we need. If first term = -1, this rule always produces a result of -1.\(A_5\), too, will = -1.

Answer B

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