the sum of the first 5 terms of a geometric series is 242, the sum of the first 10 terms is 59048,find the first term and common ratio.
Let the first term of the geometric series be a and the common ratio be r.
The sum of the first 5 terms can be written as:
S5 = a + ar + ar^2 + ar^3 + ar^4 = 242
The sum of the first 10 terms can be written as:
S10 = a + ar + ar^2 + ar^3 + ar^4 + ar^5 + ar^6 + ar^7 + ar^8 + ar^9 = 59048
Using the formula for the sum of a geometric series:
S10 = a(r^10 - 1)/(r - 1)
We can substitute the value of S10 and solve for r:
59048 = a(r^10 - 1)/(r - 1)
We are also given that:
242 = a(r^5 - 1)/(r - 1)
We can solve these two equations simultaneously to find the values of a and r. The first term a is 2 and the common ratio r is 2.