If the common ratio of an infinite G P be less than 1/2, show that each term will be greater - Maths - Sequences and Series

Question

If the common ratio of an infinite G P be less than 1/2, show that
each term will be greater than the sum of all the terms that
follows it

Mayank Jha · Accepted Answer

Dear student,

Let us represent the G P as :

a, ar2, ar3, ar4, ar5, 
Where, a = First term , r = common ratioAccording to the question,r < 12Sum of infinite G
P
 = a1 - rThe denominator (1-r) < (-12)Now, let us take ar² as the first term, thenSum = ar²1 - r, since  (1-r) < (-12)Sum < a Similarly if we take the first term as ar3, then sum < ar²And so on

Hence, we can say that each term is greater then sum of all the
terms that follows it

Regards

If the common ratio of an infinite G.P be less than 1/2, show that each term will be greater than the sum of all the terms that follows it.