Question

Which is the seventh number in the sequence 5, 9, 16, 29, 54,

• A. 300
• B. 200
• C. 330
• D. 103

Answer 1

The Correct Option is B

Given sequence: 5, 9, 16, 29, 54, ___,
To find the next numbers in the sequence, let's look at the pattern:
- Starting with 5, we notice the following:
 - \(5 \times 2 - 1 = 9\)
 - \(9 \times 2 - 2 = 16\)
 - \(16 \times 2 - 3 = 29\)
 - \(29 \times 2 - 4 = 54\)
Following this pattern, we continue:
- \(54 \times 2 - 5 = 103\)
So, the next number in the sequence is 103.
Continuing with the same pattern for the next number:
- \(103 \times 2 - 6 = 200\)
Thus, the sequence continues as: 5, 9, 16, 29, 54, 103, 200.
So the correct answer is:
Option B: 200

Answer 2

The given sequence is \( 5, 9, 16, 29, 54, \ldots \) and each term can in fact be written in closed form, which lets us evaluate every option directly instead of extending the recurrence term by term.

Writing \( a_n \) for the \( n \)-th term (so \( a_1 = 5, a_2 = 9, a_3 = 16, a_4 = 29, a_5 = 54 \)), the ratio of consecutive gaps roughly doubles each time, which suggests a solution of the form \( a_n = C \cdot 2^{n-1} + n + 1 \). Substituting \( a_1 = 5 \) gives \( C + 2 = 5 \), so \( C = 3 \), and the closed form becomes:

\( a_n = 3 \cdot 2^{n-1} + n + 1 \)

Checking this against the known terms: \( a_2 = 3(2) + 3 = 9 \), \( a_3 = 3(4) + 4 = 16 \), \( a_4 = 3(8) + 5 = 29 \), \( a_5 = 3(16) + 6 = 54 \), all match, so the formula is confirmed.

1. Option A (300): Using the formula for \( n = 7 \), \( a_7 = 3 \cdot 2^{6} + 8 = 192 + 8 = 200 \), not \( 300 \), so this option does not fit.
2. Option B (200): \( a_7 = 3 \cdot 2^{6} + 8 = 192 + 8 = 200 \), which matches exactly.
3. Option C (330): This does not equal \( a_7 = 200 \), so it is incorrect.
4. Option D (103): This is actually the sixth term, \( a_6 = 3 \cdot 2^{5} + 7 = 96 + 7 = 103 \), one term too early to be the seventh, so this is incorrect.

The seventh term of the sequence is therefore \( 200 \).

the correct answer is Option B: 200.