Question

1. in a geometric sequence, $a_3 = 651264$ and $a_4 = 162816$. find $a_{16}$ and $s_{16}$. round to 4 decimal places.

Explanation:

Step1: Find common ratio $r$

For a geometric sequence, $r = \frac{a_4}{a_3}$
$r = \frac{162816}{651264} = 0.25$

Step2: Find first term $a_1$

Use $a_n = a_1 r^{n-1}$, so $a_3 = a_1 r^{2}$
$a_1 = \frac{a_3}{r^2} = \frac{651264}{(0.25)^2} = \frac{651264}{0.0625} = 10420224$

Step3: Calculate $a_{16}$

Use $a_n = a_1 r^{n-1}$
$a_{16} = 10420224 \times (0.25)^{15}$
$(0.25)^{15} = \frac{1}{4^{15}} = \frac{1}{1073741824}$
$a_{16} = \frac{10420224}{1073741824} \approx 0.0097$

Step4: Calculate $S_{16}$

Use sum formula $S_n = a_1 \frac{1-r^n}{1-r}$
$S_{16} = 10420224 \times \frac{1-(0.25)^{16}}{1-0.25}$
$(0.25)^{16} = \frac{1}{4194304}$
$1-(0.25)^{16} \approx 0.9999997616$
$S_{16} = 10420224 \times \frac{0.9999997616}{0.75} \approx 10420224 \times 1.333333015 \approx 13893631.9997$

Answer:

$a_{16} \approx 0.0097$, $S_{16} \approx 13893631.9997$