Question

What is the nth term for 3, 10, 29, 66 and 127?

Answer 1

Therefore, the 6th term of the sequence is 108.

Explanation:

To find the pattern and the nth term for the sequence 3, 10, 29, 66, 127, we can look at the differences between the terms:

7, 19, 37, 61

The second differences are constant, which suggests that the original sequence is a polynomial of degree 2 (a quadratic). To find the formula for this quadratic, we can use the method of finite differences.

The first differences are:

7, 19, 37, 61

The second differences are:

12, 18, 24

The third differences are:

6, 6

Since the third differences are constant, we can assume that the formula for the sequence is of the form an^3 + bn^2 + cn + d, where a, b, c, and d are constants.

Using the first four terms of the sequence, we can set up a system of equations:

a(1)^3 + b(1)^2 + c(1) + d = 3

a(2)^3 + b(2)^2 + c(2) + d = 10

a(3)^3 + b(3)^2 + c(3) + d = 29

a(4)^3 + b(4)^2 + c(4) + d = 66

Solving this system of equations, we get:

a = 1/2

b = 3/2

c = 5/2

d = 1

Therefore, the formula for the nth term of the sequence is:

an^3 + bn^2 + cn + d = (1/2)n^3 + (3/2)n^2 + (5/2)n + 1

So, the nth term for the sequence 3, 10, 29, 66, 127 is:

an^3 + bn^2 + cn + d = (1/2)n^3 + (3/2)n^2 + (5/2)n + 1

To find the 6th term of the sequence, we can substitute n=6 in the formula:

a6 = (1/2)(6)^3 + (3/2)(6)^2 + (5/2)(6) + 1

= 108

Therefore, the 6th term of the sequence is 108.

Answer 2

3, 2, 7, 14, 23.

To find the nth term of a sequence, we need to identify the pattern or rule that generates the sequence.

Looking at the given sequence 3, 10, 29, 66, 127, we can see that each term is obtained by squaring the previous term and subtracting a constant value.

Specifically, we can see that:

10 = 3^2 + 1

29 = 10^2 + 9

66 = 29^2 + 7

127 = 66^2 + 1

So the pattern seems to be that each term is obtained by squaring the previous term and adding a constant value that increases by 2 for each successive term.

To find the nth term, we can use the formula:

nth term = (n-1)^2 + c

where n is the position of the term we want to find, and c is the constant value we add to the square of the previous term.

Using this formula, we can find the nth term as follows:

For the 1st term (n=1), we have 3 as the first term.

For the 2nd term (n=2), we have:

nth term = (n-1)^2 + c

= (2-1)^2 + 1

= 1 + 1

= 2

For the 3rd term (n=3), we have:

nth term = (n-1)^2 + c

= (3-1)^2 + 3

= 4 + 3

= 7

For the 4th term (n=4), we have:

nth term = (n-1)^2 + c

= (4-1)^2 + 5

= 9 + 5

= 14

For the 5th term (n=5), we have:

nth term = (n-1)^2 + c

= (5-1)^2 + 7

= 16 + 7

= 23

Therefore, the nth term for the given sequence is:

3, 2, 7, 14, 23.

hope this helps!