The next two terms of the given sequence 1 3 7 15 31
The Correct Answer and Explanation is:
Correct Answer:
The next two terms in the sequence 1, 3, 7, 15, 31 are 63 and 127.
Explanation:
To identify the next terms in a number sequence, we look for a pattern in how the terms change from one to the next.
Let us examine the given sequence:
- 1
- 3
- 7
- 15
- 31
We can look at the difference between each pair of consecutive terms:
- 3 − 1 = 2
- 7 − 3 = 4
- 15 − 7 = 8
- 31 − 15 = 16
Now observe that the differences are:
- 2 = 2¹
- 4 = 2²
- 8 = 2³
- 16 = 2⁴
This tells us that each term is formed by adding the next power of 2 to the previous term.
So to find the next number:
- Next difference should be 2⁵ = 32
- 31 + 32 = 63
Then:
- Next difference should be 2⁶ = 64
- 63 + 64 = 127
So the complete sequence now becomes:
1, 3, 7, 15, 31, 63, 127
This is a classic pattern where each term follows the formula:
aₙ = 2ⁿ − 1
Let’s confirm that:
- a₁ = 2¹ − 1 = 1
- a₂ = 2² − 1 = 3
- a₃ = 2³ − 1 = 7
- a₄ = 2⁴ − 1 = 15
- a₅ = 2⁵ − 1 = 31
- a₆ = 2⁶ − 1 = 63
- a₇ = 2⁷ − 1 = 127
This shows that the sequence is built by subtracting one from successive powers of two. The pattern is consistent and predictable.
