Question

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93. Is it better to be paid $5 per day for 100 days or to be paid 5¢ the first day,
10e the second day, 20¢ the third day, and have your salary increase in this
fashion every day for 100 days?

Answer 1

We need to find which payment method gives us more money at the end of 100 days.

It is better to be paid by the payment method where there is an increase of  cents by 2 times in salary everyday.

First payment method

Earning $5 dollars per day for 100 days.

So, total amount at the end of 100 days is $5\times 100=\ 500$

Second payment method

Earning 10 cents on day 1 and an additional 10 cents after each day.

So,

a = First element of sequence = 0.05$

r = Common ratio = $\dfrac{0.10}{0.05}=\dfrac{0.20}{0.10}=2$

n = Number of terms = 100

$S_n=\dfrac{a(r^n-1)}{r-1}\\ =\dfrac{0.05(2^{100}-1)}{2-1}\\ =\ 6.38\times 10^{28}$

So, in the second payment method an $\ 6.38\times 10^{28}$ is earned.

Hence, the second payment method is better.

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