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Question
Which of the following sequences is arithmetic progressions. For is arithmetic progression, find out the common difference.
a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2), ...
Solution
a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2), ...
Here,
First term (a) = a + b
`a_1 = ( a + 1) + b`
`a_1 = ( a + 1) + (b +1)`
Now, for the given to sequence to be an A.P,
Common difference (d) ` = a_1 - a = a_2 -a_1`
Here,
`a_1 -a = a + 1 + b - a- b`
= 1
Also,
`a_2 - a_1 = a+1+b+1-a-1-b`
= 1
Since `a_1 - a = a_2 -a_1`
Hence, the given sequence is an A.P and its common difference is d = 1
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