the last two tems in a geometric series are 1080 and 6480 and the sum ofthe series is 7775 what is the first term in the series
To find the first term of a geometric series, we need to use the given information.
Let's denote the first term of the geometric series as 'a' and the common ratio as 'r'.
We know that the last two terms of the series are 1080 and 6480. Therefore, the second-to-last term (before 6480) will be:
a * r = 1080 (Equation 1)
And the last term is:
a * r^2 = 6480 (Equation 2)
Additionally, we are given that the sum of the series is 7775. The sum of a geometric series is given by the formula:
Sum = a * (1 - r^n) / (1 - r)
where 'n' is the number of terms in the series. Since we don't know 'n', we need to find it using the given information.
To find 'n', we can use the fact that the last term divided by the second-to-last term gives us the common ratio:
6480 / 1080 = r^2 / r
6 = r
Now, let's substitute this value of 'r' into Equation 1:
a * 6 = 1080
a = 1080 / 6 = 180
Therefore, the first term in the geometric series is 180.