we have the expression
[tex]\sum_{x\mathop{=}5}^93^{(x-2)}=3^{(5-2)}+3^{(6-2)}+3^{(7-2)}+3^{(8-2)}+3^{(9-2)}[/tex][tex]\begin{gathered} =3^3+3^4+3^5+3^6+3^7 \\ =27+81+243+729+2,187 \\ =3,267 \end{gathered}[/tex]
the answer is 3,267
Another way
we have the formula
[tex]S_n=\frac{a_1-a_n*r}{1-r}[/tex]
where
For x=5
a1=3^(5-2)=27
For x=9
a5=3^(9-2)=2,187
r=3
substitute the given values in the formula
[tex]\begin{gathered} S_n=\frac{27-2,187*3}{1-3} \\ \\ S_n=3,267 \end{gathered}[/tex]
The answer is 3,267
Explanation
we have the formula
[tex]S_{n}=\frac{a_{1}-a_{n}r}{1-r}[/tex]
step 1
Find out the first term a1
a1=3^(x-2)
the first term is for x=5
a1=3^(5-2)=3^3=27
step 2
Find out the last term
an=3^(x-2)
For x=9
an=3^(9-2)=3^7=2,187
step 3
In the formula, the value of r (common ratio in geometric series) is equal to
r=3
step 3
Substitute the given values in step 1 and step 2 in the formula
[tex]\begin{gathered} S_{n}=\frac{27-2,187\times3}{1-3} \\ S_{n}=3,267 \end{gathered}[/tex]
