Consider the numerical sequence: 5, 11, 24, 51, 106, 217, \dots. What is the value of the … — State PSC Exam

Consider the numerical sequence: $5, 11, 24, 51, 106, 217, \dots$ . What is the value of the next term in this sequence?

Answer: 440

The next term in the sequence is 440.

The sequence follows the recurrence pattern where each term is twice the previous term plus an increasing integer offset:

5 \times 2 + 1 = 11

11 \times 2 + 2 = 24

24 \times 2 + 3 = 51

51 \times 2 + 4 = 106

106 \times 2 + 5 = 217

. Following this pattern, the next term is

217 \times 2 + 6 = 440

Examine the relationship between successive terms.

The differences between successive terms are

6, 13, 27, 55, 111

, and the second-level differences are

7, 14, 28, 56

(doubling each step). Alternatively, each term is multiplied by 2 and increased by an incrementing integer.

Analyzing both direct operations and higher-order differences establishes the recurrence rule

a_{n+1} = 2a_n + n

Determine the required operation for the next step in the sequence.

Multiply the 6th term (217) by 2 and add 6.

The additive component increases by 1 at each term (

+1, +2, +3, +4, +5 \rightarrow +6

Calculate the value of the next term.

217 \times 2 + 6 = 440

217 \times 2 = 434

, and adding 6 yields 440.

Recurrent series with multiplicative factor and sequential linear increment

Estimated Time:1m 30s

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