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