Question 5
The value of $$(3^{4/3}-3^{1/3})^3+(3^{5/3}-3^{2/3})^3+(3^{6/3}-3^{3/3})^3+\cdots+(3^{10/3}-3^{7/3})^3$$ is
Solution
The general term is $$(3^{(j+3)/3}-3^{j/3})^3=(2\mathbin{\times}3^{j/3})^3=8\mathbin{\times}3^j$$ for $$j=1,2,\ldots,7$$. Therefore, the sum is $$8(3+3^2+\cdots+3^7)$$. Using the geometric-series formula gives $$8\mathbin{\times}\frac{3(3^7-1)}{2}=12(3^7-1)$$.
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