Question

given the first term and the common ratio of a geometric sequence find the explicit formula.

1. ( a_1 = 2 ), ( r = 6 )

( a_n = 2 \times 6^{n - 1} )

1. ( a_1 = -4 ), ( r = 4 )

Explanation:

Step1: Recall geometric sequence formula

The explicit formula for a geometric sequence is \( a_n = a_1 \cdot r^{n - 1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step2: Substitute given values

For problem 4, we have \( a_1=-4 \) and \( r = 4 \). Substitute these into the formula: \( a_n=-4\cdot4^{n - 1} \). We can also simplify this using exponent rules: \( -4\cdot4^{n - 1}=-4^{1+(n - 1)}=-4^{n} \), but the form \( a_n=-4\cdot4^{n - 1} \) is also correct as the explicit formula.

Answer:

For \( a_1 = - 4 \), \( r = 4 \), the explicit formula is \( a_n=-4\times4^{n - 1} \) (or simplified as \( a_n=-4^{n} \))