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Question

How many three digit numbers are divisible by 11? Also find the middle term.


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Solution

Step 1: Identify the given sequence in an arithmetic progression

The first three digit number divisible by 11 is 110

The last three digit number divisible by 11is 990

The sequence of numbers 110,121,132,......,990 which is divisible by 11 is an arithmetic progression with 1st term 110 and common difference of 11

Step 2: Find the number of terms in the arithmetic progression

The nth term of an arithmetic progression is given by

an=a+(n-1)d

Here, an=990,a=110,d=11

990=110+(n-1)11880=11n-1111n=891n=81

Step 3: Find the middle term

Since the sequence has 81 terms, the 41st term is the middle term

a41=a+(41-1)da41=110+(41-1)11a41=110+40×11a41=550

Hence, there are 81 three digit numbers divisible by 11and the middle term of the sequence is 550.


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