Question 2 (iv)
Verify that each of the following is an AP and then write its next three terms.
a + b, (a + 1) + b, (a + 1) + (b + 1) . . . .

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Solution

$H e r e, a_{1} = a + b, a_{2} = (a + 1) + b, a_{3} = (a + 1) + (b + 1)$
$a_{2} - a_{1} = (a + 1) + b - (a + b) = a + 1 + b - a = 1$
$a_{3} - a_{2} = (a + 1) + (b + 1) - [(a + 1) + b]$
$= a + 1 + b + 1 - a - 1 - b = 1$
$∵ a_{2} - a_{1} = a_{3} - a_{2} = 1 = C o m m o n d i f f e r e n c e$
Since the difference between successive terms are same, it forms an AP.
The next three terms are,
$a_{4} = a_{1} + 3 d = a + b + 3 (1) = (a + 2) + (b + 1)$
$a_{5} = a_{1} + 4 d = a + b + 4 (1) = (a + 2) + (b + 2)$
$a_{6} = a_{1} + 5 d = a + b + 5 (1) = (a + 3) + (b + 2)$

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