What is the nth term of the sequence 7, 15, 31, 63, 127, …?
Answer and explanation
Correct answer: 2ⁿ⁺² − 1
The governing concept is recognizing an exponential sequence and expressing it explicitly. Rewrite the terms as 8 − 1, 16 − 1, 32 − 1, 64 − 1, and 128 − 1. The powers of 2 are 2³, 2⁴, 2⁵, 2⁶, and 2⁷, so for the nth term the power is 2ⁿ⁺². Therefore aₙ = 2ⁿ⁺² − 1, which is option A. Checking n = 1 gives 2³ − 1 = 7, and n = 5 gives 2⁷ − 1 = 127. Option B is shifted by one power, option C does not preserve the doubling pattern, and option D is linear rather than exponential.
Frequently asked questions
What is the correct answer to this question?
2ⁿ⁺² − 1
Why is this the correct answer?
The governing concept is recognizing an exponential sequence and expressing it explicitly. Rewrite the terms as 8 − 1, 16 − 1, 32 − 1, 64 − 1, and 128 − 1. The powers of 2 are 2³, 2⁴, 2⁵, 2⁶, and 2⁷, so for the nth term the power is 2ⁿ⁺². Therefore aₙ = 2ⁿ⁺² − 1, which is option A. Checking n = 1 gives 2³ − 1 = 7, and n = 5 gives 2⁷ − 1 = 127. Option B is shifted by one power, option C does not preserve the doubling pattern, and option D is linear rather than exponential.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: Explicit or general rule.
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