Question

find the sum of the first 10 terms of the following sequence. round to the nearest hundredth if necessary.
21,84,336,...
○ 7,000,525.00
○ 7,340,025.00
○ 1,048,575.00
○ 6,990,000.00

Explanation:

Step1: Identify sequence type

Check if the sequence is geometric. Divide consecutive terms: $\frac{84}{21} = 4$, $\frac{336}{84} = 4$. So it's a geometric sequence with first term $a_1 = 21$ and common ratio $r = 4$.

Step2: Use geometric series sum formula

The formula for the sum of the first $n$ terms of a geometric series is $S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$ (for $r
eq 1$). Here, $n = 10$, $a_1 = 21$, $r = 4$.

Step3: Calculate the sum

Substitute values into the formula:
$S_{10} = 21 \cdot \frac{4^{10} - 1}{4 - 1}$
First, calculate $4^{10} = 1048576$.
Then, $4^{10} - 1 = 1048575$.
Next, $\frac{1048575}{3} = 349525$.
Finally, $S_{10} = 21 \cdot 349525 = 7340025$.

Answer:

7,340,025.00 (corresponding to the option "7,340,025.00")