Introduction to sum of arithmetic sequence:
Arithmetic sequence or progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 10, 20, 30, 40, 50… is an arithmetic sequence with common difference d = 5. Two types of sequences are arithmetic sequence and geometric sequence.
Formula – Sum of Arithmetic Sequence
The formula for the nth term an of an infinite arithmetic sequence with a common difference d and a first term a1 is given by
an = a1 + (n - 1 )d
The sum sn of the first n terms of an infinite arithmetic sequence is defined by
sn = a1 + a2 + a3 + ... + a n
and is a1 is given by
sn = n (a1 + an) / 2
Examples – Sum of Arithmetic Sequence
Ex 1: Find the sum of all positive integers, from 27 to 4590 inclusive, which are divisible by 27.
Sol : Sequence of first few numbers of positive integers divisible by 27 are given by 27, 54, 81...
The above sequence has a first number equal to 27 and a common difference d = 27.
We need to know the rank of the term 4590.
We use the following formula for the nth term
an = a1 + (n - 1 )d
4590 = a1 + (n - 1 )d
Substitute a1 and d by their values
4590 = 27 + 27(n - 1)
Solve for n to obtain
n = 170
4590 is the 170th term, we can use the following formula to find sum
sn = n (a1 + an) / 2
s170 = 170 (27 + 4590) / 2 = 392445.
Ex 2: Find the sum of all positive integers, from 23 to 3680 inclusive, which are divisible by 23.
Sol : Sequence of first few numbers of positive integers divisible by 23 are given by 23, 46, 69...
The above sequence has a first number equal to 23 and a common difference d = 23.
We need to know the rank of the term 3680.
We use the following formula for the nth term
an = a1 + (n - 1 )d
3680 = a1 + (n - 1 )d
Substitute a1 and d by their values
3680 = 23 + 23(n - 1)
Solve for n to obtain
n = 160
3680 is the 160th term, we can use the following formula to find sum
sn = n (a1 + an) / 2
s160 = 160 (23 + 3680) / 2 = 296240.
Arithmetic sequence or progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 10, 20, 30, 40, 50… is an arithmetic sequence with common difference d = 5. Two types of sequences are arithmetic sequence and geometric sequence.
Formula – Sum of Arithmetic Sequence
The formula for the nth term an of an infinite arithmetic sequence with a common difference d and a first term a1 is given by
an = a1 + (n - 1 )d
The sum sn of the first n terms of an infinite arithmetic sequence is defined by
sn = a1 + a2 + a3 + ... + a n
and is a1 is given by
sn = n (a1 + an) / 2
Examples – Sum of Arithmetic Sequence
Ex 1: Find the sum of all positive integers, from 27 to 4590 inclusive, which are divisible by 27.
Sol : Sequence of first few numbers of positive integers divisible by 27 are given by 27, 54, 81...
The above sequence has a first number equal to 27 and a common difference d = 27.
We need to know the rank of the term 4590.
We use the following formula for the nth term
an = a1 + (n - 1 )d
4590 = a1 + (n - 1 )d
Substitute a1 and d by their values
4590 = 27 + 27(n - 1)
Solve for n to obtain
n = 170
4590 is the 170th term, we can use the following formula to find sum
sn = n (a1 + an) / 2
s170 = 170 (27 + 4590) / 2 = 392445.
Ex 2: Find the sum of all positive integers, from 23 to 3680 inclusive, which are divisible by 23.
Sol : Sequence of first few numbers of positive integers divisible by 23 are given by 23, 46, 69...
The above sequence has a first number equal to 23 and a common difference d = 23.
We need to know the rank of the term 3680.
We use the following formula for the nth term
an = a1 + (n - 1 )d
3680 = a1 + (n - 1 )d
Substitute a1 and d by their values
3680 = 23 + 23(n - 1)
Solve for n to obtain
n = 160
3680 is the 160th term, we can use the following formula to find sum
sn = n (a1 + an) / 2
s160 = 160 (23 + 3680) / 2 = 296240.
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