Maths is this questions answer below?

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1 Answer
Steve M
Dec 13, 2017

# sum_(n=64)^98 \ (4n^2-5) = 932645 #

Making (b) the correct solution.

Explanation:

We seek:

# S = sum_(n=64)^98 \ (4n^2-5) #
# S = sum_(n=1)^98 \ (4n^2-5) - sum_(n=1)^63 \ (4n^2-5) #

Using the standard summation formulas:

# sum_(r=1)^n \ r^2 = 1/6n(n+1)(2n+1) #

We have:

# sum_(r=1)^n \ (4r^2-5) = 4sum_(r=1)^n r^2 - sum_(r=1)^n 5 #

# " " = 2/3 n(n+1)(2n+1) - 5n #

# " " = n/3{ 2(n+1)(2n+1) - 15} #

# " " = n/3{ 2(2n^2+3n+1) - 15} #

# " " = n/3{ 4n^2+6n+2 - 15} #

# " " = n/3{ 4n^2+6n-13} #

Using this result we have:

# S_98 = 98/3{ 38416+588-13} #
# \ \ \ \ \ = 98/3{ 38991 } #
# \ \ \ \ \ = 1273706 #

And:

# S_63 = 63/3{ 15876 + 378 - 13} #
# \ \ \ \ \ = 63/3{ 16241 } #
# \ \ \ \ \ = 341061 #

So then:

# S = S_98 - S_63 #
# \ \ = 1273706 - 341061 #
# \ \ = 932645 #

Making (b) the correct solution.

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