The mean of 10 numbers \(7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14, \dots\) is ___________
Show Hint
For sequences defined as products of APs, always expand the general term into a polynomial in \(n\). This allows the direct use of standard summation formulae for \(n, n^2, \dots\)
Step 1: Understanding the Concept:
We first identify the general term (\(a_n\)) of the sequence, calculate the sum of the first 10 terms using sigma notation, and then divide by 10 to find the mean. Step 2: Detailed Explanation:
The sequence is \(7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14, \dots\)
1st factors: \(7, 10, 13, \dots\) (AP with \(a=7, d=3\)). General term = \(7 + (n-1)3 = 3n + 4\).
2nd factors: \(8, 10, 12, \dots\) (AP with \(a=8, d=2\)). General term = \(8 + (n-1)2 = 2n + 6\).
General term \(a_n = (3n+4)(2n+6) = 6n^2 + 18n + 8n + 24 = 6n^2 + 26n + 24\).
Total sum \(S_{10} = \sum_{n=1}^{10} (6n^2 + 26n + 24)\).
\[ S_{10} = 6 \sum n^2 + 26 \sum n + \sum 24 \]
\[ \sum_{n=1}^{10} n^2 = \frac{10(11)(21)}{6} = 385 \]
\[ \sum_{n=1}^{10} n = \frac{10 \times 11}{2} = 55 \]
\[ S_{10} = 6(385) + 26(55) + 24 \times 10 = 2310 + 1430 + 240 = 3980 \]
Mean = \(S_{10} / 10 = 3980 / 10 = 398\). Step 3: Final Answer:
The mean is 398.
