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Question

1.    classify as arithmetic, geometric, or neither. Finally find an explicit formula (rule for the nth term) for each.  Designate the first term with n = 1.

Answer 1

Step 1:

The series is .

The arithmetic series have the common difference.

The geometric series have the common ratio.

The above series is in the form  of arithmetic series.

Arithmetic series is .

Where is first term

            is common difference.

Common difference .

Where is term .

            is term .

term in arithmetic series .

Step 2:

Now compare the above equation with arithmetic series.

The first term in the given series is .

Common difference .

term in arithmetic series .

.

Solution :

The series is in the form  of arithmetic series.

Answer 2

(b)

Step 1:

The series is .

The arithmetic series have the common difference.

The geometric series have the common ratio.

The above series is in the form  of arithmetic series.

Arithmetic series is .

Where is first term

            is common difference.

Common difference .

Where is term .

            is term .

term in arithmetic series .

Step 2:

Now compare the above equation with arithmetic series.

The first term in the given series is .

Common difference .

term in arithmetic series .

.

Solution :

The series is in the form  of arithmetic series.

 

Answer 3

(c)

Step 1:

The series is .

The above series have the common constant ratio.

So above series is in the form  of geometric series.

geometric series :

.

Where is first term

            is common ration.

Common ration .

Where is term .

            is term .

term in geometric series .

Step 2:

Now compare the above equation with geometric series.

The first term in the given series is .

Common ratio .

term in geometric series .

.

Solution :

The series is in the form  of geometric series.

Answer 4

(d)

Step 1:

The series is .

The above series have the common constant ratio.

So above series is in the form  of geometric series.

geometric series :

.

Where is first term

            is common ratio.

Common ratio .

Where is term .

            is term .

term in geometric series .

Step 2:

.

Now compare the above equation with geometric series.

The first term in the given series is .

Common ratio .

In the given series is the term.

But term in geometric series .

As base are equal , equate powers.

So ,the series has terms .

Solution :

The series is in the form  of geometric series.

Answer 5

(e)

Step 1:

The series is .

The arithmetic series have the common difference.

The geometric series have the common ratio.

But the above series does not have common difference as well as common ratio.

We can notice that the series is made up by the squares of consecutive numbers.

Rewrite the series

So the term in the series is .

.

Solution :

The series is neither arithmetic series nor geometric series.