Questions: Find the sum of the first n terms of the sequence. 5,20,80, ... ; n=9

Solution Steps

The given sequence is \( 5, 20, 80, \ldots \). We observe that each term is obtained by multiplying the previous term by a common ratio.

To find the common ratio \( r \), we divide the second term by the first term: \[ r = \frac{20}{5} = 4 \]

The formula for the sum of the first \( n \) terms of a geometric sequence is given by: \[ S_n = a \frac{r^n - 1}{r - 1} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Substituting the known values \( a = 5 \), \( r = 4 \), and \( n = 9 \) into the formula: \[ S_9 = 5 \frac{4^9 - 1}{4 - 1} \]

Now, we compute \( S_9 \) using the values substituted: \[ S_9 = 5 \frac{4^9 - 1}{3} \] Calculating \( 4^9 \) and then finding \( S_9 \) gives us the sum of the first 9 terms of the sequence.

Final Answer