Thomas Ray's parents begin saving to buy their son a car for his 16th birthday. They save $800 the first year and each year they save 5% more than the previous year. How much money will they have saved for his 16th birthday? (I'm in the arithmetic and geometric series + sigma notation part of my lessons)

Answer: they have saved $18925.99 for his 16th birthday.

Step-by-step explanation:

We know that they save $800 per year, and in each year after the first, they add a 5% extra (0.05 in decimal form).

then, the first year the amount is $800.

the second year, they add $800 + 0.05*$800 = $800*1.05

the third year, they add: $800*1.05 + 0.05*$800*1.05 = $800 * (1.05) ^2

Now is easy to see that the relation is:

C (n) = $800 * (1.05) ^ (n)

where n goes from 0 to 15, and represents the 16 years in which the parents are saving money.

now, we know that for a geometric series we have:

∑a*r^n = a * (1 + r^N) / (1 + r)

where the sumation goes from 0 to N - 1.

in our case, N - 1 = 15, so N = 16. a = $800 and r = 1.05

then the total of money is;

T = $800 * (1 - 1.05^16) / (1 - 1.05) = $18925.99