The sum of the first two terms of an infinite G.P. is 5, and each term is three times the sum of the succeeding terms. Find the G.P.

Suppose the 1st term is a and the common ratio is r.


we can say that GP looks like: a ,ar ,ar2 ,…


According to question:


a + ar = 5 …equation 1


Also, a1 = 3(a2+a3+a4+…∞) {you can take any other combination}


a = 3(ar+ar2+ar3 + …∞)


1 = 3(r + r2 + r3 +…∞)


We observe that above progression possess a common ratio. So it is a geometric progression.


Common ratio = r and first term (a) = r


Sum of infinite GP = ,where a is the first term and k is the common ratio.


Note: We can only use the above formula if |k|<1


we can use the formula for the sum of infinite GP.


Therefore



1-r = 3r


r =


From equation 1:


a+ar = a(1+r) = 5.


So,



a = 4


GP is (4 , 1 , 1/4 , 1/16 , ….)


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