Question

The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary).
9, comma, 15, comma, 25, comma, point, point, .
9,15,25,...
Find the 10th term.
Find the 10th term..

Answer 1

To find the 10th term of the sequence given the first three terms: 9, 15, and 25, we first need to determine the pattern of the sequence.

First, we can calculate the differences between consecutive terms:

1. The difference between the second term (15) and the first term (9) is:
\[
15 - 9 = 6
\]

2. The difference between the third term (25) and the second term (15) is:
\[
25 - 15 = 10
\]

So, the sequence of terms we have so far is:
- 1st term: 9
- 2nd term: 15
- 3rd term: 25

Let's look at the differences:
- The difference between the 1st and 2nd terms is 6.
- The difference between the 2nd and 3rd terms is 10.

Now, let's find the differences of these differences (second differences):
- The difference between the first difference (6) and the second difference (10):
\[
10 - 6 = 4
\]

Since the second difference is constant, this is a quadratic sequence. The general form of a quadratic sequence can be expressed as:
\[
a_n = An^2 + Bn + C
\]

Where:
- \( A \) is half of the second difference.
- \( B \) can be determined using the known terms.
- \( C \) is the first term when \( n=1 \).

Since the second difference is \( 4 \):
\[
A = \frac{4}{2} = 2
\]

Now let's write the sequence in terms of \( A \):
\[
a_n = 2n^2 + Bn + C
\]

We know:
- \( a_1 = 9 \) => \( 2(1)^2 + B(1) + C = 9 \)
- \( a_2 = 15 \) => \( 2(2)^2 + B(2) + C = 15 \)
- \( a_3 = 25 \) => \( 2(3)^2 + B(3) + C = 25 \)

From \( a_1 \):
\[
2 + B + C = 9 \quad \text{(1)}
\]
From \( a_2 \):
\[
8 + 2B + C = 15 \quad \text{(2)}
\]
From \( a_3 \):
\[
18 + 3B + C = 25 \quad \text{(3)}
\]

Solving equations (1), (2), and (3):
From (1):
\[
B + C = 7 \quad \text{(1)}
\]
From (2):
\[
2B + C = 7 \quad \text{(2)}
\]

Subtract equation (1) from equation (2):
\[
(2B + C) - (B + C) = 7 - 7
\]
\[
B = 0
\]

Now plug \( B \) back into equation (1):
\[
0 + C = 7 \quad \Rightarrow \quad C = 7
\]

Then the expression for the nth term becomes:
\[
a_n = 2n^2 + 0 \cdot n + 7 = 2n^2 + 7
\]

Now to find the 10th term:
\[
a_{10} = 2(10)^2 + 7 = 2(100) + 7 = 200 + 7 = 207
\]

Thus, the 10th term of the sequence is:
\[
\boxed{207}
\]