Answer:
The sum of the first 6 terms is 3,412.5.
Step-by-step explanation:
The second term of the geometric series is given by:
[tex]a_{2}=a_{1}*r[/tex]
Where a1 is the first term and r is the common ratio. The seventh term can be written as a function of the second term as follows:
[tex]a_{7}=a_{1}*r^{6} \\a_{7}=a_{2}*r^{5} \\10,240 = 10*r^{5}\\r=\sqrt[5]{1024} \\r = 4[/tex]
The sum of "n" terms of a geometric series is given by:
[tex]a_{1} = \frac{10}{4} = 2.5\\S_{n}=a_{1}(\frac{r^{n}-1 }{r-1})\\S_{6}=2.5(\frac{4^{6}-1 }{4-1})\\S_{6}=3,412.5[/tex]
The sum of the first 6 terms is 3,412.5.
The sum of the first six terms of the geometric series with second term as 10 and seventh term as 10240 is; 3412.5
We are given;
Second term of geometric series; a₂ = 10
Seventh term of geometric series; a₇ = 10240
In geometric series, we know that;
a₂/a₁ = r
a₁ = a₂/r
where r is the common ratio.
Also, formula for 7th term will be;
a₇ = a₁ * r⁶
Thus;
a₇ = (a₂/r) * r⁶
a₇ = a₂ * r⁵
10240 = 10r⁵
r = [tex]\sqrt[5]{\frac{10240}{10} }[/tex]
r = 4
Thus;
a₁ = 10/4
a₁ = 2.5
Formula for sum of n terms of a geometric series is;
Sₙ = a₁(rⁿ - 1)/(r - 1)
S₆ = 2.5(4⁶ - 1)/(4 - 1)
S₆ = 3412.5
Read more about Geometric series at; https://brainly.com/question/2515230
