Questions: ∑(j=0)^3 6^j =

Solution Steps

The given expression is a finite geometric series of the form: \[ \sum_{j=0}^{n} ar^{j} \] where \( a = 1 \) (since \( 6^{j} \) can be written as \( 1 \cdot 6^{j} \)), \( r = 6 \), and \( n = 3 \).

The sum of a finite geometric series is given by: \[ \sum_{j=0}^{n} ar^{j} = a \cdot \frac{r^{n+1} - 1}{r - 1} \] Substitute \( a = 1 \), \( r = 6 \), and \( n = 3 \) into the formula: \[ \sum_{j=0}^{3} 6^{j} = 1 \cdot \frac{6^{3+1} - 1}{6 - 1} \]

Calculate \( 6^{4} \): \[ 6^{4} = 6 \cdot 6 \cdot 6 \cdot 6 = 1296 \] Substitute this back into the formula: \[ \sum_{j=0}^{3} 6^{j} = \frac{1296 - 1}{5} = \frac{1295}{5} \]

Divide \( 1295 \) by \( 5 \): \[ \frac{1295}{5} = 259 \]

Final Answer