Question

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 1

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

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

Answer 2

$18,925.99

Explanation:

The sum of n=16 terms of the geometric series with first term a1=800 and common ratio r=1.05 will be ...

Sn = a1·(r^n -1)/(r -1)

S16 = $800·(1.05^16 -1)/(1.05 -1) ≈ $18,925.99