Questions: Estou preocupada com os meus gastos: a cada més que passa, a minha reserva económica, inicialmente de R25600,00, fica reduzida à metade. Quantos meses decorrerâo ate que ela se torne R 400,00 ?

Estou preocupada com os meus gastos: a cada més que passa, a minha reserva económica, inicialmente de R25600,00, fica reduzida à metade. Quantos meses decorrerâo ate que ela se torne R 400,00 ?

Transcript text: Estou preocupada com os meus gastos: a cada més que passa, a minha reserva económica, inicialmente de R\$25600,00, fica reduzida à metade. Quantos meses decorrerâo ate que ela se torne R\$ 400,00 ?

Solution

Solution Steps

To solve this problem, we need to determine how many months it will take for the initial amount of R$25600.00 to be reduced to R$400.00 if it is halved each month. This can be approached by repeatedly dividing the amount by 2 until it is less than or equal to R$400.00, while counting the number of divisions (months).

Step 1: Define the Problem

We need to find the number of months it takes for an initial amount of $ R\$25600.00 $ to be reduced to $ R\$400.00 $ by halving the amount each month.

Step 2: Set Up the Equation

The problem can be expressed as a geometric sequence where the initial term $ a_0 = 25600 $ and each subsequent term is half of the previous term. We need to find the smallest integer $ n $ such that:

\[ a_n = \frac{25600}{2^n} \leq 400 \]

Step 3: Solve the Inequality

To find $ n $, solve the inequality:

\[ \frac{25600}{2^n} \leq 400 \]

Multiply both sides by $ 2^n $:

\[ 25600 \leq 400 \times 2^n \]

Divide both sides by 400:

\[ 64 \leq 2^n \]

Step 4: Calculate $ n $

Find the smallest $ n $ such that $ 2^n \geq 64 $. Since $ 2^6 = 64 $, we have:

\[ n = 6 \]