Sponsor Area

Permutations And Combinations

Question

Evaluate:

$space space sum from straight i equals 1 to 10 of open square brackets open parentheses 1 half close parentheses to the power of straight i minus 1 end exponent plus open parentheses 1 fifth close parentheses to the power of straight i plus 1 end exponent close square brackets$

Solution

space space space space sum from straight i space equals space 1 to 10 of open square brackets open parentheses 1 half close parentheses to the power of straight i minus 1 end exponent plus space open parentheses 1 fifth close parentheses to the power of straight i plus 1 end exponent close square brackets space equals space open square brackets open parentheses 1 half close parentheses to the power of 0 space plus space open parentheses 1 fifth close parentheses squared close square brackets space plus space open square brackets open parentheses 1 half close parentheses to the power of 1 plus open parentheses 1 fifth close parentheses cubed close square brackets

plus open square brackets open parentheses 1 half close parentheses squared space plus space open parentheses 1 fifth close parentheses to the power of 4 close square brackets space plus space........ space plus space open square brackets open parentheses 1 half close parentheses to the power of 9 space plus space open parentheses 1 fifth close parentheses to the power of 11 close square brackets

equals space open square brackets 1 plus 1 half plus 1 over 2 squared plus..... plus space 1 over 2 to the power of 9 close square brackets space plus space open square brackets 1 over 5 squared plus 1 over 5 cubed plus 1 over 5 to the power of 4 plus...... plus space 1 over 5 to the power of 11 close square brackets

WiredFaculty

2 open parentheses 1 minus 1 over 2 to the power of 10 close parentheses space plus space 1 over 20 open parentheses 1 minus 1 over 5 to the power of 10 close parentheses