Question

Find n,if 242 is the nth term of 2,4,10,14.....

Answer 1

In the given sequence, the difference between consecutive terms is not constant. To find a pattern, let's look at the differences between the terms:

1st difference:
4 - 2 = 2

2nd difference:
10 - 4 = 6

3rd difference:
14 - 10 = 4

It appears that the differences between the differences are constant. Therefore, it suggests that the sequence has a quadratic pattern.

Let's use the formula for the nth term of a quadratic sequence:

nth term = an^2 + bn + c

To find the values of a, b, and c, we can substitute the first three terms of the sequence into the formula:

2 = a(1)^2 + b(1) + c
4 = a(2)^2 + b(2) + c
10 = a(3)^2 + b(3) + c

Simplifying these equations, we get:

a + b + c = 2
4a + 2b + c = 4
9a + 3b + c = 10

Solving this system of equations, we find that a = 2, b = -2, and c = 2.

Therefore, the formula for the nth term is:

nth term = 2n^2 - 2n + 2

Now, let's substitute 242 into this formula and solve for n:

242 = 2n^2 - 2n + 2

Simplifying this equation, we get:

2n^2 - 2n - 240 = 0

Dividing through by 2, we have:

n^2 - n - 120 = 0

Factoring this quadratic equation, we get:

(n - 12)(n + 10) = 0

Thus, n = 12 or n = -10.

Since n represents the position of the term in the sequence, it cannot be negative. Therefore, the value of n is 12.