Arithmetic Progression

1. Arithmetic
Progression
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2. Arithmetic Progression
a)5, 8, 11, 14, 17, 20, … 3n+2, …
b)-4, 1, 6, 11, 16, … 5n – 9, . . .
c)11, 7, 3, -1, -5, … -4n + 15, . . .
In all the lists above, we see that successive terms
are obtained by adding a fixed number to the preceding
terms. Such list of numbers is said to form an Arithmetic
Progression ( AP ).
So,an arithmetic progression is a list of numbers in which
each term is obtained by adding a fixed number to the
preceding term except the first term.
This fixed number is called the common difference of the AP.
Remember that it can be positive, negative or zero
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3. n
th
term of arithmetic sequence
T
n
= a + d(n – 1)
a = First term
d = common difference
n = number of terms.
Common difference = the difference between two consecutive
terms in a sequence.
d = T
n
– T
n-1
Example
Find the n
th
term of the following AP.
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4. Finding the 956
th
term
56, 140, 124, 108, . . .
T
n
= a + d(n – 1)
T
956
= 156 + -16(956 – 1)
T
956
= 156 - 16(955)
T
956
= 156 - 15280
T
956
= -15124
a
1
= 156
d = -16
n = 956
Example
Finding the number of terms in the AP
10, 8, 6, 4, 2, . . .-24
T
n
= a + d(n – 1)
-24 = 10 -2(n – 1)
-34 = -2(n – 1)
17 = n-1
n = 18
a = 10
d = -2
T
n
= -24
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5. The 5
th
term of an AP is 13 and the 13
th
term is -19. Find
the first term & the common difference.
T
5
= a + 4d = 13……..(1)
T
13
= a + 12d = -19……….(2)
(2) – (1): 8d = -19 - 13
8d = - 32

d = -4
Substitute d = -4 into (1):
a + 4(-4) = 13
a – 16 = 13
a = 29
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6. S
n
= a
1
+ (a
1
+ d) + (a
1
+ 2d) + …+ a
n
S
n
= a
n
+ (a
n
- d) + (a
n
- 2d) + …+ a
1
2
)(
1
1
n
n
i
in
aan
aS
+
==å
=
)(2
1 nn aanS +=
)(...)()()(2
1111 nnnnn
aaaaaaaaS ++++++++=
Sum of First terms of an AP
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7. 1 + 4 + 7 + 10 + 13 + 16 + 19
a
1
= 1
a
n
= 19
n = 7
2
)(
1 n
n
aan
S
+
=
2
)191(7+
=
nS
2
)20(7
=
nS
70=
n
S
Example
Find the sum of the integers from 1 to 100
a
1
= 1
a
n
= 100
n = 100
2
)(
1 n
n
aan
S
+
=
2
)1001(100+
=
nS
2
)101(100
=
nS
5050=
nS
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8. Find the sum of the multiples of 3
between 9 and 1344
a
1
= 9
a
n
= 1344
d = 3
2
)(
1 n
n
aan
S
+
=
2
)13449(+
=
n
S
n
2
)1353(446
=
nS
301719=
n
S
)1(
1 -+= ndaa
n
)1(391344 -+= n
3391344 -+= n
631344 +=n
n31338=
n=446
S
n
= 9 + 12 + 15 + . . . + 1344
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9. Find the sum of the multiples of 7
between 25 and 989
a
1
= 28
a
n
= 987
d = 7
2
)(
1 n
n
aan
S
+
=
2
)98728(+
=
n
S
n
2
)1015(138
=
nS
70035=
n
S
)1(
1
-+= ndaa
n
)1(728987 -+= n
7728987 -+= n
217987 +=n
n7966=
n=138
S
n
= 28 + 35 + 42 + . . . + 987
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10. Evaluate
a
1
= 16
a
n
= 82
d = 3
n = 23
2
)(
1 n
n
aan
S
+
=
2
)8216(23+
=
nS
2
)98(23
=
nS
1127=
n
S
S
n
= 16 + 19 + 22 + . . . + 82
å
=
+
25
3
)73(
i
i
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11. Review -- Arithmetic
n
th
term Sum of n terms
)1(
1
-+= ndaa
n
2
)(
1 n
n
aan
S
+
=
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12. The End
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