Geometric series are $\sum_{n=1}^{4}5(2)^{n-1}$, $\sum_{n=1}^{5}3(4)^{n-1}$, \(\sum_{n=1}^{3}10(0.5)^{n-1}

Question

Geometric series are $\sum_{n=1}^{4}5(2)^{n-1}$, $\sum_{n=1}^{5}3(4)^{n-1}$, $\sum_{n=1}^{3}10(0.5)^{n-1}$. The expanded forms are 10+5+2.5, 3+12+48+192+768, 5+10+20+40.

GPT-5 mini · Answer

Those expansions are correct but listed in a different order. Using a = first term and r = ratio, S_n = a(1 − r^n)/(1 − r):

- ∑_{n=1}^{4} 5(2)^{n-1} = 5 + 10 + 20 + 40 = 75.
- ∑_{n=1}^{5} 3(4)^{n-1} = 3 + 12 + 48 + 192 + 768 = 1023.
- ∑_{n=1}^{3} 10(0.5)^{n-1} = 10 + 5 + 2.5 = 17.5.

Geometric series are \(\sum_{n=1}^{4}5(2)^{n-1}\), \(\sum_{n=1}^{5}3(4)^{n-1}\), \(\sum_{n=1}^{3}10(0.5)^{n-1}\). The expanded forms are 10+5+2.5, 3+12+48+192+768, 5+10+20+40.

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