Find the next term in the following number sequence: 3, 5, 8, 13, 22, 39, 72, \dots — State PSC Exam

Find the next term in the following number sequence:

3, 5, 8, 13, 22, 39, 72, \dots

Answer: 137

The next number in the given series is 137.

Computing second-level differences between terms reveals a doubling sequence (

1, 2, 4, 8, 16

). The next second-level difference is

32

. Adding

32

to the last first-level difference (

33

) yields

65

. Finally, adding

65

to the last term (

72

) gives the correct value of

137

Compute the first-order differences between consecutive terms in the series.

The first-order differences are

2, 3, 5, 9, 17, 33

Analyzing differences helps determine whether the growth rate follows an arithmetic, geometric, or polynomial rule.

Compute the second-order differences of the first-order difference sequence.

The second-order differences are

1, 2, 4, 8, 16

Because the first-order differences do not show an obvious linear pattern, evaluating higher-order differences is necessary.

Identify the underlying rule governing the second-order differences.

The second-order sequence doubles with each step (

2^0, 2^1, 2^2, 2^3, 2^4, \dots

). The next second-order difference is

16 \times 2 = 32

Recognizing geometric doubling (

2^n

) allows for projecting the next step accurately.

Reconstruct the next first-order difference and calculate the required next sequence value.

Next first-order difference

= 33 + 32 = 65

. Next series term

= 72 + 65 = 137

Working backward through the difference layers yields the exact value of the next term.

Double-Difference Series with Geometric Progression

Estimated Time:2m 0s

Question