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Question

If a1,a2,a3,..are in a harmonic progression with a1=5 and a20=25. The least positive integer n for which an<0is


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Solution

Step 1. Finding The least positive integer n for which an<0

Given a1,a2,a3,...are in HP

1a1,1a2,1a3are in AP

And a1=5 and a20=25

1a1=15 and 1a20=125

Let d be the common difference,

Step 2. Find the common difference:

As we know,

an=a+(n-1)d

1a20=1a1+19d

1a201a1=19d

125-15=19d

-425=19d

d=-425×119

an<0

a+(n-1)d<0

15+(n-1)-425×119<0

4(n-1)19×5>1

n-1>19×54

n>19×54+1

n=25

Hence, The least positive integer n for which an<0is 25


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