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Bot The formula for the sum of n terms of a GP is:
Sn = a(1 - r^n) / (1 - r)
where a is the first term of the GP and r is the common ratio.
We are given that r = 4/5 and Sn = 125/9, so we can substitute those values into the formula and solve for a:
125/9 = a(1 - (4/5)^n) / (1 - 4/5)
125/9 = a(1 - (4/5)^n) / (1/5)
125/9 = 5a(1 - (4/5)^n)
Dividing both sides by 5 gives:
25/9 = a(1 - (4/5)^n)
We know that the first term of the GP must be positive, so we can take the limit as n approaches infinity to find the smallest possible value for a:
lim (4/5)^n = 0
So, we have:
25/9 = a(1 - 0)
a = 25/9
Therefore, the first term of the GP is 25/9.