Alternating sums

Alternating sum of powers of negative one ●○○○○ 10 ptsFind $(-1)^{-10} + (-1)^{-9} + (-1)^{-8} + \cdots + (-1)^9 + (-1)^{10}$ . (The sum has 21 terms, one for each integer exponent from $-10$ to $10$ .)
Alternating sum of consecutive integers ●○○○○ 10 ptsWhat is the value of the expression $100 - 99 + 98 - 97 + 96 - 95 + \cdots + 4 - 3 + 2 - 1$ ?
Alternating sum of integers up to 2007 ●○○○○ 10 ptsFind the value of $1 - 2 + 3 - 4 + \cdots + 2005 - 2006 + 2007$ .
Alternating sum of a linear sequence ●●○○○ 20 ptsThe $n$ th term of a sequence is $a_n = (-1)^{n+1}(3n + 2)$ . What is the value of $a_1 + a_2 + \dots + a_{100}$ ?
Alternating block sum of squares ●●○○○ 20 ptsCompute $100^{2}+99^{2}-98^{2}-97^{2}+96^{2}+95^{2}-94^{2}-93^{2}+\ldots+4^{2}+3^{2}-2^{2}-1^{2}$ .
Alternating sum in steps of four ●●○○○ 20 ptsFind the value of $2013-2009+2005-2001+\cdots+29-25$ .
Sum of alternate terms of a sequence ●●●○○ 50 ptsA sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?
Counting alternating-parity increasing sequences ●●●○○ 50 ptsA finite strictly increasing sequence of integers is called alternating if its first term is odd, its second term is even, its third term is odd, and so on, alternating in parity. The empty sequence also counts as alternating. Let $A(n)$ be the number of alternating sequences whose terms are all taken from $\{1,2,\ldots,n\}$ , so $A(1)=2$ and $A(2)=3$ . Find $A(20)$ .
Alternating sum of minimum values ●●●○○ 50 ptsFor a positive integer $n$ , define $f_n(x)=\left(\sum_{i=1}^{n}|x-i|\right)^2-\sum_{i=1}^{n}(x-i)^2$ on the interval $[0,n+1]$ . Let $a_n$ be the minimum value of $f_n(x)$ . Find $\sum_{n=1}^{11}(-1)^{n+1}a_n$ .

More in Finite sums

Comparing sums Pairing terms Sums of 1 to n, odds and evens Sums of consecutive integers Sums of cubes and powers Sums of multiples Sums of products Telescoping products Terms from partial sums