Find the value of the expression $\left (2 + 5 \right ) + \left (2^{2}+5^{2} \right ) + \left (2^{3}+5^{3} \right ) + \left (2^{4}+5^{4} \right )$

Question

How to effectively solve this expression?

$$\left (2 + 5 \right ) + \left (2^{2}+5^{2} \right ) + \left (2^{3}+5^{3} \right ) + \left (2^{4}+5^{4} \right )$$

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Inefficient method:

$$\left (7 \right ) + \left (29 \right ) + \left (133 \right ) + \left (641 \right ) = 810$$

Answer 1

It's $$(2+2^2+2^3+2^4)+(5+5^2+5^3+5^4) = (1+2+ \dots+2^4)+(1+5+ \dots +5^4)-2 =$$ $$=(2^5-1)+ (5^5-1)/4 -2 = 31 + 781 - 2 = 810$$

Answer 2

Use the geometric sum: $\sum_{k=1}^{n}m^k=\frac{m^{n+1}-1}{m-1}-1$

Answer 3

$$\left (2 + 5 \right ) + \left (2^{2}+5^{2} \right ) + \left (2^{3}+5^{3} \right ) + \left (2^{4}+5^{4} \right ) = \left(\sum_{i=1}^4 2^i\right)+\left(\sum_{i=1}^4 5^i\right) = \frac{2-2^5}{1-2} + \frac{5-5^5}{1-5} = 30 + 5(5^4-1)/4 = 30+5\times 624/4=30+780=810$$