To determine the 10th term of the sequence given by the first three terms \(9, 15, 25\), we first need to identify the pattern in the sequence.
Let's analyze the differences between the consecutive terms:
-
The difference between the second term and the first term: \[ 15 - 9 = 6 \]
-
The difference between the third term and the second term: \[ 25 - 15 = 10 \]
Now, let's look at the differences of the differences:
- The difference between \(10\) and \(6\): \[ 10 - 6 = 4 \]
Now we see that the first set of differences is \(6\) and \(10\), and the second set of differences is \(4\). This suggests that the differences are increasing linearly.
Assuming the sequence follows a quadratic formula \(a_n = an^2 + bn + c\), where \(n\) is the term number, we can find the coefficients \(a\), \(b\), and \(c\) using the initial terms:
-
For \(n = 1\): \[ a(1^2) + b(1) + c = 9 \implies a + b + c = 9 \tag{1} \]
-
For \(n = 2\): \[ a(2^2) + b(2) + c = 15 \implies 4a + 2b + c = 15 \tag{2} \]
-
For \(n = 3\): \[ a(3^2) + b(3) + c = 25 \implies 9a + 3b + c = 25 \tag{3} \]
Now we can solve these equations step by step.
Subtract equation (1) from equation (2): \[ (4a + 2b + c) - (a + b + c) = 15 - 9 \] \[ 3a + b = 6 \tag{4} \]
Subtract equation (2) from equation (3): \[ (9a + 3b + c) - (4a + 2b + c) = 25 - 15 \] \[ 5a + b = 10 \tag{5} \]
Now subtract equation (4) from equation (5): \[ (5a + b) - (3a + b) = 10 - 6 \] \[ 2a = 4 \implies a = 2 \]
Now substitute \(a = 2\) back into equation (4): \[ 3(2) + b = 6 \implies 6 + b = 6 \implies b = 0 \]
Finally, substitute \(a = 2\) and \(b = 0\) back into equation (1): \[ 2 + 0 + c = 9 \implies c = 7 \]
So, the formula for the \(n\)-th term of the sequence is: \[ a_n = 2n^2 + 7 \]
Now, we can find the 10th term \(a_{10}\): \[ a_{10} = 2(10^2) + 7 = 2(100) + 7 = 200 + 7 = 207 \]
Thus, the 10th term of the sequence is: \[ \boxed{207} \]
4 answers