Question

Find the nth term rule for the sequence: 6, 14, 24, 36, ...

Answer 1

n² + 5n is the nth term rule for the sequence

Explanation:

6 14 24 36...

8 10 12 (first difference)

2 2 (second difference)

second difference = 2

now,

here the nth term would be an² + bn + c

Now,

second difference = 2a

or, 2 = 2a

so, a = 1

To solve for b,

if n = 2,

an² + bn + c = 14

or, 2² + b×2 + c = 14

or, 4 + 2b + c = 14

so, 2b + c = 10

so, c = 10 - 2b

Now,

if n = 1,

an² + bn +c = 6

1² + b × 1 + c = 6

or, 1 + b + c = 6

or, b+c = 5

as we know,

c = 10-2b

Now,

b+c = 5

or, b - (10 - 2b) = 5

so, b - 10 + 2b = 5

so, 3b = 15

so, b = 5

now,

c = 10-2b

or, c = 10 - 2×5

so, c = 10-10 = 0

so

an² + bn + c = n² + 5n + 0

so, the nth term = n² + 5n