Question

use the formula \\(\sum_{i=0}^{n} ar^i = a \cdot \frac{1 - r^{n+1}}{1 - r}\\) to evaluate the sum \\(\sum_{i=0}^{9} 10 \cdot 3^i\\).

Explanation:

Step1: Identify values for formula

From $\sum_{i=0}^{9} 10 \cdot 3^i$, we get $a=10$, $r=3$, $n=9$.

Step2: Substitute into sum formula

$$ \sum_{i=0}^{n} ar^i = a \cdot \frac{1 - r^{n+1}}{1 - r} $$

Substitute values:

$$ 10 \cdot \frac{1 - 3^{9+1}}{1 - 3} $$

Step3: Simplify exponent and denominator

$$ 10 \cdot \frac{1 - 3^{10}}{-2} $$

Calculate $3^{10}=59049$:

$$ 10 \cdot \frac{1 - 59049}{-2} $$

Step4: Simplify numerator

$$ 10 \cdot \frac{-59048}{-2} $$

Step5: Compute final value

$$ 10 \cdot 29524 = 295240 $$

Answer:

295240