Math, asked by ASHIISH53861, 1 year ago

Derive The Formula Of Finding nth term from the end of an AP

Answers

Answered by shi22052004
0

Answer:

#MarkAsBrainliest

Step-by-step explanation:

Consider the following AP:

2

,

5

,

8

,

11

,

13...

The first term  

a

of this AP is 2, the second term is 5, the third term is 8, and so on. We write this as follows:

T

1

=

a

=

2

T

2

=

5

T

3

=

8

.

.

.

The  

n

t

h

term of this AP will be denoted by  

T

n

. How can you find the  

n

t

h

term for any value of  

n

? For example, what will be the value of the following terms?

T

20

,

T

45

,

T

90

,

T

200

Obviously, we cannot evaluate each and every term of the AP to determine these specific terms. Instead, we must develop a relation which enables us to find the  

n

t

h

term for any value of  

n

.

To do that, consider the following relations for the terms in an AP:

T

1

=

a

T

2

=

a

+

d

T

3

=

a

+

d

+

d

=

a

+

2

d

T

4

=

a

+

2

d

+

d

=

a

+

3

d

T

5

=

a

+

3

d

+

d

=

a

+

4

d

T

6

=

a

+

4

d

+

d

=

a

+

5

d

.

.

.

✍Note: Please go through Arithmetic Progressions to understand about First Term  

(

a

)

and Common Difference  

(

d

)

in a better way.

What pattern do you observe? If you have to calculate the sixth term, for example, then you have to add five times  

d

to the first term  

a

. Similarly, if you have to calculate the  

n

t

h

term, how many times will you have to add  

d

to  

a

? The answer should be easy: one less than  

n

. Thus,

T

n

=

a

+

(

n

1

)

d

✍Note: This relation helps us calculate any term of an AP, given its first term and its common difference.

Thus, for the AP above, we have:

T

20

=

2

+

(

20

1

)

3

=

2

+

57

=

59

T

45

=

2

+

(

45

1

)

3

=

2

+

132

=

134

T

90

=

2

+

(

90

1

)

3

=

2

+

267

=

269

T

200

=

2

+

(

200

1

)

3

=

2

+

597

=

599

Answered by varadad25
2

Answer:

Formula for the nth term of an A.P. :

tn = a + ( n - 1 ) d

Step - by - step explanation:

Generally, in the A. P. t1, t2, t3,...... if the first term is 'a' and the common difference is 'd',

t1 = a

t2 = t1 + d = a + d = a + ( 2 - 1 ) d

t3 = t2 + d = a + d + d = a + 2d = a + ( 3 - 1 ) d

t4 = t3 + d = a + 2d + d = a + 3d = a + ( 4 - 1 ) d

∴ we get the formula

tn = a + ( n - 1 ) d

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