Question

The sum $1^2-2 \cdot 3^2+3 \cdot 5^2-4 \cdot 7^2+5 \cdot 9^2-\ldots+15 \cdot 29^2$ is_____

Answer 1

Correct Answer: 6952

To solve the sum \(1^2-2 \cdot 3^2+3 \cdot 5^2-4 \cdot 7^2+\ldots+15 \cdot 29^2\), observe the pattern of the series. Each term is defined as \((-1)^{n+1}n\cdot (2n-1)^2\) for \(n=1\) to \(15\). Simplifying each term, the sum is calculated as follows:

1. Generate each term using: \(T_n=(-1)^{n+1}\cdot n(2n-1)^2\). 

2. Compute each individual term:

n	Term \(T_n\)
1	\((-1)^2\cdot 1 \cdot (2\cdot 1-1)^2 = 1 \cdot 1^2 = 1\)
2	\((-1)^3\cdot 2 \cdot (2\cdot 2-1)^2 = -2 \cdot 3^2 = -18\)
3	\((-1)^4\cdot 3 \cdot (2\cdot 3-1)^2 = 3 \cdot 5^2 = 75\)
4	\((-1)^5\cdot 4 \cdot (2\cdot 4-1)^2 = -4 \cdot 7^2 = -196\)
5	\((-1)^6\cdot 5 \cdot (2\cdot 5-1)^2 = 5 \cdot 9^2 = 405\)
6	\((-1)^7\cdot 6 \cdot (2\cdot 6-1)^2 = -6 \cdot 11^2 = -726\)
7	\((-1)^8\cdot 7 \cdot (2\cdot 7-1)^2 = 7 \cdot 13^2 = 1,183\)
8	\((-1)^9\cdot 8 \cdot (2\cdot 8-1)^2 = -8 \cdot 15^2 = -1,800\)
9	\((-1)^{10}\cdot 9 \cdot (2\cdot 9-1)^2 = 9 \cdot 17^2 = 2,601\)
10	\((-1)^{11}\cdot 10 \cdot (2\cdot 10-1)^2 = -10 \cdot 19^2 = -3,610\)
11	\((-1)^{12}\cdot 11 \cdot (2\cdot 11-1)^2 = 11 \cdot 21^2 = 4,851\)
12	\((-1)^{13}\cdot 12 \cdot (2\cdot 12-1)^2 = -12 \cdot 23^2 = -6,348\)
13	\((-1)^{14}\cdot 13 \cdot (2\cdot 13-1)^2 = 13 \cdot 25^2 = 8,125\)
14	\((-1)^{15}\cdot 14 \cdot (2\cdot 14-1)^2 = -14 \cdot 27^2 = -10,206\)
15	\((-1)^{16}\cdot 15 \cdot (2\cdot 15-1)^2 = 15 \cdot 29^2 = 12,615\)

3. Sum all terms: \(1 - 18 + 75 - 196 + 405 - 726 + 1,183 - 1,800 + 2,601 - 3,610 + 4,851 - 6,348 + 8,125 - 10,206 + 12,615 = 6,952\).

The computed sum is 6,952, which fits within the expected range.