Question

find the value of the following expression and round to the nearest integer:
\\(\sum_{n=2}^{98} 600(1.04)^{n - 2}\\)

Explanation:

Step1: Rewrite the summation index

Let $k = n-2$. When $n=2$, $k=0$; when $n=98$, $k=96$.
The sum becomes $\sum_{k=0}^{96} 600(1.04)^k$

Step2: Identify geometric series parameters

This is a geometric series with first term $a=600$, common ratio $r=1.04$, number of terms $m=97$.

Step3: Apply geometric series formula

The sum of a geometric series is $S = a\frac{r^m - 1}{r - 1}$
Substitute values:
$S = 600\times\frac{(1.04)^{97} - 1}{1.04 - 1}$

Step4: Calculate $(1.04)^{97}$

Using a calculator: $(1.04)^{97} \approx 56.0848$

Step5: Compute numerator and denominator

Numerator: $56.0848 - 1 = 55.0848$
Denominator: $1.04 - 1 = 0.04$

Step6: Compute the fraction

$\frac{55.0848}{0.04} = 1377.12$

Step7: Multiply by initial term

$600\times1377.12 = 826272$

Answer:

826272