Question

What comes next in the sequence; 1, 6, 13, 22, 33, ?

• A. \(35 \)
• B. \(38 \)
• C. \(46 \)
• D. \(49 \)

Hint:

When the primary differences between numbers form an arithmetic progression (\(5, 7, 9, 11, \dots\)), it means the terms are increasing by consecutive odd numbers. Keep an eye out for this classic quadratic sequence structure!

Answer 1

The Correct Option is C

Concept: To solve number sequence problems, we must analyze the regular mathematical relationship between consecutive terms. A standard and highly reliable method is looking at the common difference between each term and its immediate successor to check for a uniform progression.

Step 1: Calculate the differences between consecutive terms.
Let the terms of the given sequence be represented as \( T_1, T_2, T_3, T_4, T_5, T_6 \). We are given: \[ T_1 = 1, \quad T_2 = 6, \quad T_3 = 13, \quad T_4 = 22, \quad T_5 = 33 \] Let us find the difference between each consecutive pair: \[\begin{aligned} \text{First difference } (D_1) &= T_2 - T_1 = 6 - 1 = 5 \\ \text{Second difference } (D_2) &= T_3 - T_2 = 13 - 6 = 7 \\ \text{Third difference } (D_3) &= T_4 - T_2 = 22 - 13 = 9 \\ \text{Fourth difference } (D_4) &= T_5 - T_3 = 33 - 22 = 11 \end{aligned}\]

Step 2: Recognize the underlying pattern among the calculated differences.
Let us collect the list of computed differences: \[ 5, \, 7, \, 9, \, 11 \] As we can clearly observe, the differences themselves are consecutive odd integers starting from 5. The pattern shows that each consecutive difference grows steadily by a constant value of \(+2\): \[ 7 - 5 = 2 \] \[ 9 - 7 = 2 \] \[ 11 - 9 = 2 \] This indicates that the sequence possesses a constant second-order difference, confirming a quadratic underlying behavior.

Step 3: Extrapolate the pattern to compute the next term (\(T_6\)).
Following the established progression of adding \(+2\) to the previous difference, the next consecutive difference (\(D_5\)) must logically be: \[ D_5 = 11 + 2 = 13 \] Therefore, to arrive at the sixth term (\(T_6\)), we simply add this difference value of 13 to the fifth term (\(T_5 = 33\)): \[ T_6 = T_5 + D_5 \] \[ T_6 = 33 + 13 = 46 \] Thus, the next missing number in this sequence is 46, matching Option (C).