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:
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The difference between the second term (15) and the first term (9) is: \[ 15 - 9 = 6 \]
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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: \[ \boxed{207} \]
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