3 consider the sequence: $left{40, -8, \frac{8}{5}, -\frac{8}{25}, \\dots\\right}$\nwrite its explicit…

3 consider the sequence: $left{40, -8, \frac{8}{5}, -\frac{8}{25}, \\dots\\right}$\nwrite its explicit formula and use it to find the 11th term.

3 consider the sequence: $left{40, -8, \frac{8}{5}, -\frac{8}{25}, \\dots\\right}$\nwrite its explicit formula and use it to find the 11th term.

Answer

Explanation:

Step1: Identify common ratio $r$

$r = \frac{-8}{40} = -\frac{1}{5}$

Step2: State geometric sequence formula

The explicit formula for a geometric sequence is $a_n = a_1 r^{n-1}$, where $a_1=40$, $r=-\frac{1}{5}$. Substitute values: $a_n = 40 \left(-\frac{1}{5}\right)^{n-1}$

Step3: Simplify the explicit formula

$a_n = 40 \cdot \frac{(-1)^{n-1}}{5^{n-1}} = 40 \cdot 5^{1-n} (-1)^{n-1} = 8 \cdot 5 \cdot 5^{1-n} (-1)^{n-1} = 8(-1)^{n-1}5^{2-n}$

Step4: Calculate 11th term ($n=11$)

$a_{11} = 40 [LLM SSE On Failure]