S.5 Mathematics
Summation of A.S. and G.S.
S.5 Mathematics (Summation of A.S. and G. S.) Worksheet 1
＜Arithmetic Series＞
An arithmetic series is the indicated sum of the terms of the corresponding arithmetic
sequence.
e.g. 2  5  8  11   is an arithmetic series corresponding to the arithmetic sequence
2, 5, 8, 11,  .
Sum to n Terms of Arithmetic Series
Let S(n) be the sum to n terms and l be the last term (i.e. the nth term) of an
arithmetic series, then
or
Notes : (1)
(2)
e.g. (i)
S ( n) 
n
(a  l )
2
……………….(1)
S ( n) 
n
[2a  (n  1)d ]
2
……………….(2)
S 1  T 1  a
T n  S n  S n  1
If the 1st term and the 12th term of an arithmetic series are  7 and 37
respectively, then the sum to 12 terms is
S (12) 
(ii) If the 1st term and the common difference of an arithmetic series are –2 and
–3 respectively, then the sum to 12 terms is
S (12) =
Example 1
Find the sum of the first 20 terms of an arithmetic sequence
Example 2
Find the sum of the arithmetic series
3, 2, 7, …… .
2 + 9 + 16 + ……+ 100.
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S.5 Mathematics
Summation of A.S. and G.S.
Example 3
Given the first term and the kth term of an arithmetic sequence are 20 and 54 respectively. The
sum of the first k term is 999, find
(a) the value of k and the common difference,
(b) the sum from the 21st term to 40th term.
Example 4
If the 5th term of an arithmetic sequence is 50 and the sum of the 3rd and 6th term is 50, find the
sum of the first 10 terms.
Example 5
In an A.S., S 4  3 and the sum from the 6th term to the 11th term is 85.5. Find the sum of the
first 15 terms of this sequence.
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S.5 Mathematics
Summation of A.S. and G.S.
Example 6
Given S n  2n 2  7n , find the first three terms of this sequence.
Example 7
(a) Find the number of multiples of 2 from 12 to 250 inclusive, and hence determine the sum of
all the multiples of 2 from 12 to 250 inclusive.
(b) Hence, or otherwise, find the sum of all the multiples of 2 from 12 to 250 inclusive,
excluding those common multiples of 2 and 5.
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S.5 Mathematics
Summation of A.S. and G.S.
S.5 Mathematics (Summation of A.S. and G. S.) Worksheet 2
＜Geometric Series＞
A geometric series is the indicated sum of the terms of the corresponding geometric
sequence.
e.g. 3  6 12  24   is a geometric series corresponding to the geometric sequence
3, 6, 12, 24,  .
Sum to n Terms of Geometric Series
If S (n ) denotes the sum to n terms of a geometric series where r  1 , then
S ( n) 
Notes :
a(1  r n )
1 r
………………(3)
(1) If r  1 , then S (n)  na .
(2) S 1  T 1  a
(3) T n  S n  S n  1
Example 1
Find the sum of the first 10 terms of the geometric sequence
18, 9,
9
, …… .
2
Example 2
(a) Find the total number of terms in the G.S. : 3, 6, 12, 24, ……, 6144.
(b) Hence, find the sum of all the negative terms in the above sequence.
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S.5 Mathematics
Summation of A.S. and G.S.
Example 3
If the sum of the first n terms of the geometric: 3, 9, 27, 81, … is smaller than 9840, find
the minimum value of n.
Example 4
David deposits $P in a bank which pays an interest at a rate of R % per annum, calculated on a
monthly basis. At the end of each month, he withdraws
1
of the amount in the account
4
(including principal and interest) and keeps the remainder to be redeposited at the same rate.
Let $Tn be the sum of money that David withdraws at the end of nth month.
(a) Express T1, T2, T3 in terms of P and R.
(b) T1, T2, T3 , … form a geometric sequence. Express the common ratio in terms of R.
(c) If P = 12 000 and R = 12, find the sum of money that David withdraws for the first year.
(Give the answer correct to the nearest $1.)
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S.5 Mathematics
Summation of A.S. and G.S.
S.5 Mathematics (Summation of A.S. and G. S.) Worksheet 3
＜Sum to Infinity of Geometric Series＞
When  1  r  1 , the sum to infinity of a geometric series, denoted by S () ,
is given by:
S ( ) 
Note :
a
1 r
………………(4)
If r  1 or r  1 , S () cannot be determined.
e.g. The sum to infinity of the geometric series 24  12  6   is
S () 
Example 1
Find the sum to infinity of the geometric sequence
2 , 1,
2 1
, , …… .
2 2
Example 2
If the sum to infinity of a given geometric series is 6 times of its first term, find its common ratio.
Example 3
A ball falls from a height of 9 m and rebounds to a height of 4 m. The ball continues to fall and
rebound such that each time the height of the rebound is
4
of the height from which it
9
previously fell.
(a) What height does the ball reach after the fifth rebound?
(b) What is the total distance traveled by the ball before it comes to rest?