Math, asked by CedricDiggory, 1 year ago

find the sum to n terms: 1/1*4 + 1/4*7 + 1/7*10 ...

Answers

Answered by sairishitamann
1

Answer:

n

3

n

+

1

.

Explanation:

Let,

t

n

denote the

n

t

h

term of the series,

s

n

=

1

1

4

+

1

4

7

+

1

7

10

+

...

to n terms.

Observe that, the First Factors of the Dr. of

t

n

are

1

,

4

,

7

,

...

,

which form an A.P., with the first term

a

=

1

,

and the

common difference,

d

=

3

;

n

t

h

t

e

r

m

=

a

+

(

n

1

)

d

=

1

+

3

(

n

1

)

=

3

n

2

.

Similarly, for the Second Factors,

n

t

h

t

e

r

m

=

3

n

+

1

.

t

n

=

1

(

3

n

2

)

(

3

n

+

1

)

.

t

n

=

1

3

{

3

(

3

n

2

)

(

3

n

+

1

)

}

,

=

1

3

{

(

3

n

+

1

)

(

3

n

2

)

(

3

n

2

)

(

3

n

+

1

)

}

,

=

1

3

{

3

n

+

1

(

3

n

2

)

(

3

n

+

1

)

3

n

2

(

3

n

2

)

(

3

n

+

1

)

}

,

=

1

3

{

1

3

n

2

1

3

n

+

1

}

,

s

n

=

1

3

n

1

{

1

3

n

2

1

3

n

+

1

}

,

=

1

3

{

(

1

1

1

4

)

+

(

1

4

1

7

)

+

(

1

7

1

10

)

+

...

+

(

1

3

n

5

1

3

n

2

)

+

(

1

3

n

2

1

3

n

+

1

)

}

,

=

1

3

{

1

1

1

3

n

+

1

}

,

=

1

3

{

(

3

n

+

1

)

1

3

n

+

1

}

,

s

n

=

n

3

n

+

1

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