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# Prove that No Matter What the Real Numbers a and B Are, the Sequence with the Nth Term a + Nb is Always an A.P. What is the Common Difference? - CBSE Class 10 - Mathematics

#### Question

Prove that no matter what the real numbers a and are, the sequence with the nth term a + nb is always an A.P. What is the common difference?

#### Solution

In the given problem, we are given the sequence with the nth term (a_n) as a + nb where a and b are real numbers.

We need to show that this sequence is an A.P and then find its common difference (d)

Here,

a_n = a + nb

Now, to show that it is an A.P, we will find its few terms by substituting n = 1, 2, 3

So,

Substituting n = 1we get

a_1 = a + (1)b

a_1 = a + b

Substituting n = 2we get

a_2 = a+ (2)b

a_2 = a + 2b

Substituting n = 3we get

a_3 = a + (3)b

a_3 = a + 3b

Further, for the given to sequence to be an A.P,

Common difference (d)  = a_2 - a_1 = a_3 - a_2

Here

a_2 - a_1 = a + 2b - a - b

= b

Also

a_3- a_2 = a + 3b - a - 2b

= b

Since a_2 - a_1 = a_3 - a_2

Hence, the given sequence is an A.P and its common difference is d = b

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Solution Prove that No Matter What the Real Numbers a and B Are, the Sequence with the Nth Term a + Nb is Always an A.P. What is the Common Difference? Concept: Arithmetic Progression.
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