Product Of Geometric Sequence

Relevant wiki: Geometric Progressions

Let the common ratio be $r$ . Then we have:

$\begin{aligned} g_1 g_8 & = g_1\cdot g_1r^7 & \small \color{#3D99F6}{\text{Note that }g_n = g_1 r^{n-1}} \\ & = g_1r^2 \cdot g_1r^5 \\ & = g_3 \cdot g_6 \\ & = 10 \cdot 1000 \\ & = \boxed{10000} \end{aligned}$

Relevant wiki: Geometric Progressions

$g_3 = g_1 × q^2 \ , \ g_6 = g_1 × q^5 \ and \ g_8 = g_1 × q^7$ , where $q$ is the quotient of the geometric sequence, therefore:

$g_1 × g_8 = g_1^2 × q^7 = g_3 × g_6 = 10 ×1000 = \boxed {10000}$

For generally, if $a + b = c + d$ , then $g_a \times g_b = g_c \times g_d$ .

This can be easily shown by substituting the values $g_a = g_1 \times r ^ { a- 1 }$ into the expression.

Hence, we obtain $g_1 \times g_ 8 = g_3 \times g_6 = 10000$ .

$g_{1} \times g_{8} = g_{3} \times g_{6} = 10 \times 1000 \boxed{10000}$

Let $g_3=ar^2$ and $g_6=ar^5$ , where $a$ is first term and $r$ is common difference.

General Formula: $g_n=ar^{n-1}$

$\Rightarrow g_3= ar^2=10 \ ...(1)$
$\Rightarrow g_6= ar^5=10^3 \ ...(2)$

Now,

$\Rightarrow g_1×g_8=\color{#D61F06}{a×ar^7}$

Multiplying $(1)$ and $(2)$ .

$\Rightarrow g_3×g_6=\color{#D61F06}{ar^2×ar^5}=10000$

$\therefore g_1×g_8=\boxed{10000}$