Question

if each term of an AP is multiplied or divided by a nonzero fixed number, then the resulting is progression is AP​

Answer 1

Answer:Let {a1, a2, a3, a4, ..............}   ........... (i)  be an Arithmetic Progression with common difference d.

Again, let k be a fixed constant quantity.

Now k is added to each term of the above A.P. (i)

Then the resulting sequence is a1 + k, a2 + k, a3 + k, a4 + k ..................

Let bn = an + k, n = 1, 2, 3, 4, ............

Then the new sequence is b1, b2, b3, b4, ...............

We have bn+1 - bn = (an+1 + k) - (an + k) = an+1 - an = d for all n ∈ N, [Since, <an> is a sequence with common difference d].

Therefore, the new sequence we get after adding a constant quantity k to each term of the A.P. is also an Arithmetic Progression with common difference d.

To get the clear concept of property I let us follow the below explanation.

Let us assume ‘a’ be the first term and ‘d’ be the common difference of an Arithmetic Progression. Then, the Arithmetic Progression is {a, a + d, a + 2d, a + 3d, a + 4d, ..........}

Step-by-step explanation: