Question

use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the following sequence with the given first term, ( a_1 ), and common ratio, ( r ). find ( a_{14} ) when ( a_1 = 4 ) and ( r = -4 ). ( a_{14} = square ) (type an integer or a decimal.)

Explanation:

Step1: Recall geometric sequence formula

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

Step2: Substitute the given values

We are given \( a_1 = 4 \), \( r = - 4 \), and \( n = 14 \). Substitute these into the formula:
\( a_{14}=4\times(-4)^{14 - 1} \)

Step3: Simplify the exponent

First, calculate the exponent: \( 14 - 1 = 13 \), so we have \( a_{14}=4\times(-4)^{13} \)

Step4: Calculate \((-4)^{13}\)

Since 13 is odd, \( (-4)^{13}=-4^{13} \). And \( 4^{13}=4\times4\times\cdots\times4 \) (13 times). But we can calculate it as \( (-4)^{13}=-67108864 \)

Step5: Multiply by 4

Now, multiply by 4: \( 4\times(-67108864)=-268435456 \)

Answer:

\( - 268435456 \)