In March, a family starts saving for a vacation they are planning for the end of August. The family expects the vacation to cost $1544.

Question

In March, a family starts saving for a vacation they are planning for the end of August. The family expects the vacation to cost $1544. They start with $125 at the start of each month they plan to deposit 25% more than the previous month. Will they have enough money for their trip? If not, how much more do they need?

Answer 1

This math problem is a summation of a geometric sequence.

To find out how much money the family has saved by the final deposit made in August before they take their vacation, we need to find the sum of the 6 months by using the

Geometric Summation Formula: S_n = a₁(1-rⁿ)/(1-r)

S_n represents the sum of the finite series of n terms

a₁ represents the value of the first term to be summed (March deposit)

r represents the common ratio (1.25 since they are adding 25% each month onto the previous month)

n represents the number of terms to be summed (6 for the months of March through August)

a₁ = 125

r = 1.25

n = 6

Equation: S₆ = 125(1 - 1.25⁶)/(1 - 1.25) Simplify the equation

S₆ = 1407.35 (rounded)

The family will have $1407.35 after they make their last deposit in August.

This means they WILL NOT have enough money for their vacation.

$1544 - $1407.35 = $136.65 additional funds are required to fund the vacation.

Answer 2

a₁ = 125

r = 1.25

n = 6

S_n = a₁(rⁿ - 1) / (r - 1)