Question

You are given a sequence of numbers: 2, 5, 10, 17, ... Identify the pattern and find the next three terms in the sequence. Can you express the nth term of this sequence in a general formula?

• A. 26, 37, 50, \( n^2 + 1 \)
• B. 26, 41, 58, \( n^2 + 1 \)
• C. 26, 39, 54, \( n^2 + 1 \)
• D. 27, 42, 59, \( n^2 + 1 \)

Hint:

For quadratic sequences, the second difference between terms is constant. This allows you to derive the general form of the nth term.

Answer 1

The Correct Option is A

The given sequence is 2, 5, 10, 17, ... We can observe that the difference between successive terms is increasing: \[ 5 - 2 = 3, \quad 10 - 5 = 5, \quad 17 - 10 = 7 \] The difference increases by 2 each time. This is characteristic of a quadratic sequence. Let's assume the nth term has the form: \[ T(n) = n^2 + 1 \] Check the first few terms: For \( n = 1 \), \( T(1) = 1^2 + 1 = 2 \). For \( n = 2 \), \( T(2) = 2^2 + 1 = 5 \). For \( n = 3 \), \( T(3) = 3^2 + 1 = 10 \). For \( n = 4 \), \( T(4) = 4^2 + 1 = 17 \). Thus, the next terms will be: For \( n = 5 \), \( T(5) = 5^2 + 1 = 26 \), For \( n = 6 \), \( T(6) = 6^2 + 1 = 37 \), For \( n = 7 \), \( T(7) = 7^2 + 1 = 50 \).