Question

Oana cheltuiește o sumă de bani în trei zile. În prima zi Oana cheltuiește jumătate din sumă, a

doua zi cheltuiește jumătate din suma rămasă, iar a treia zi restul de 100 lei. Calculați suma totală
cheltuită de Oana în cele trei zile

Answer 1

700 lei

Explanation:

You have a geometric sequence for which you know the common ratio and the last term.

There are only three terms, so you can solve the problem in either of two ways.

1. The brute force method (easiest for only a few terms)

Each term is half the one before it, so each term is double the one after it.

3rd term = 100 lei

2nd term = 200

1st term = 400

Total = 700 lei

Oana spent 700 lei

2. Using formulas (best for longer sequences)

The general formula for your sequence is

aₙ = a₁rⁿ⁻¹

For your sequence,

a₃ = 100; r = 0.5

(a) Calculate a₁

Set the last term equal to the general formula.

a₃ = a₁(0.5)ⁿ⁻¹

100 = a₁(0.5)² = 0.25a₁

a₁ = 100/0.25 = 400

(b) Calculate the sum

The general formula for the sum of a geometric sequence is

`\begin{array}{rcl}S_(n)& =& (a_(1)(1 - r^(n)) )/( 1 - r)\\\\S_(3)& =& (400(1 - 0.5^(3)) )/( 1 - 0.5)\\\\& =& (400(1 - 0.125) )/( 0.5)\\\\& =& (400*0.875 )/( 0.5)\\\\& = & \mathbf{700}\\\end{array}`

Oana spent 700 lei.