Question

Which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference.

(i) 3, 6, 12, 24, ...

(ii) 0, −4, −8, −12, ...

(iii) $\frac{1}{2},\frac{1}{4},\frac{1}{6},\frac{1}{8},...$

(iv) 12, 2, −8, −18, ...

(v) 3, 3, 3, 3, ...

(vi) p, p + 90, p + 180 p + 270, ... where p = (999)⁹⁹⁹

(vii) 1.0, 1.7, 2.4, 3.1, ...

(viii) −225, −425, −625, −825, ...

(ix) 10, 10 + 2⁵, 10 + 2⁶, 10 + 2⁷,...

(x) a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2), ...

(xi) 1², 3², 5², 7², ...

(xii) 1², 5², 7², 73, ...

Answer 1

In the given problem, we are given various sequences.

We need to find out that the given sequences are an A.P or not and then find its common difference (d)

(i)

Here,

First term (a) = 3

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

Common difference (d)

Here,

Also,

Since

Hence, the given sequence is not an A.P

(ii)

Here,

First term (a) = 0

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

Common difference (d)

Here

Also,

Since

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

(iii)

Here,

First term (a) =

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

Common difference (d)

Here,

Also,

Since

Hence, the given sequence is not an A.P

(iv)

Here,

First term (a) = 12

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

Common difference (d)

Here,

Also,

Since

Hence, the given sequence is an A.P with the common difference

(v)

Here,

First term (a) = 3

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

Common difference (d)

Here,

Also,

Since

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

(vi) Where,

Here,

First term (a) = p

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

Common difference (d)

Here,

Also,

Since

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

(vii)

Here,

First term (a) = 1.0

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

Common difference (d)

Here,

Also,

Since

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

(viii)

Here,

First term (a) = −225

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

Common difference (d)

Here,

Also,

Since

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

(ix)

Here,

First term (a) = 10

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

Common difference (d) = $a_1-a=a_2-a_1$

Here,

Also,

Since

Hence, the given sequence is not an A.P

(x)

Here,

First term (a) = a + b

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

Common difference (d)

Here,

Also,

Since

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

(xi)

Here,

First term (a) =

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

Common difference (d)

Here,

Also,

Since

Hence, the given sequence is not an A.P

(xii)

Here,

First term (a) =

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

Common difference (d)

Here,

Also,

$a_3-a_2=7^2-5^2=49-25=24$

Since

Hence, the given sequence is an A.P with the common difference .