Properties of Summation

Algebra 2 · Axiom Academy

LESSON Properties of Summation Four rules that make linear — factor, split, count, and slice — each true because a sum is just repeated addition. When every term of a sum carries the same factor c , that factor belongs to the whole sum — pull it out in front of the . Each term is really c equal copies of a_i ; the animation regroups those copies into c clean copies of the whole sum , which is exactly what " c out front" means. Factor the 3 out of . The leftover sum is just 1+2+3+4+5 = 15 , so the whole thing is . For a_i = b_i = i over i = 1,2,3 : the left side is 1+4+9 = 14 , but . If each term is itself a sum a_i + b_i , the terms are interleaved : an a -part and a b -part added together. Since addition can be reordered, we may collect all the a -parts into one sum and all the b -parts into another. The animation de-interleaves one combined sum into two parallel sums running side by side. (The same works for a difference — subtract the second sum instead.) Split into its two pieces: and . 3. Add the Same Number n Times What if the summand does not depend on the index at all? Then every term is the same constant c , and the sum just adds c to itself once for each index value. The animation stacks n identical copies of c into a single tall bar — its height is . Add 7 once for each i from 1 to 6 . There are 6 copies, so the total is .

This is the written version of the interactive lesson above. See the full Algebra 2 course.