Consider the following numerical sequence: 5, 14, 19, 54, 83, \dots What is the value of t… — State PSC Exam

Consider the following numerical sequence:

5, 14, 19, 54, 83, \dots

What is the value of the next term in this sequence?

Answer: 202

The next term in the numerical sequence is 202.

The sequence follows the recurrence pattern

a_n = 2 a_{n-1} + (-1)^n n^2

starting with

a_1 = 5

. To find the 6th term, double the 5th term (

83 \times 2 = 166

) and add

6^2 = 36

, resulting in

166 + 36 = 202

Identify the operation connecting the first two terms (

5

and

14

a_2 = 2 \times a_1 + 2^2 = 2(5) + 4 = 14

Establishes the initial multiplier and square term offset.

Determine the rule for the third, fourth, and fifth terms.

a_3 = 2(14) - 3^2 = 19

a_4 = 2(19) + 4^2 = 54

a_5 = 2(54) - 5^2 = 83

Shows an alternating addition and subtraction of consecutive squares

n^2

after doubling the previous term.

Apply the rule for

n=6

to calculate the missing term.

a_6 = 2(83) + 6^2 = 166 + 36 = 202

Since the 5th term subtracted

5^2

, the 6th term must add

6^2

to twice the 5th term.

Alternating Operational and Square Offsets in Recursive Series

Question