Introduction to Infinite Arithmetic Sequence:
Arithmetic involves the rules for primary operations that are applicable to whole numbers, integers, rational and irrational numbers, in a single word, to all the real numbers. 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 7, 14, 21, 28, 35… is an arithmetic sequence with common difference d = 5.
Formula for Infinite 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
Example Problems on Infinite Arithmetic Sequence
Ex1: Find the sum of all positive integers, from 13 to 3250 inclusive, which are divisible by 13.
Sol : Sequence of first few numbers of positive integers divisible by 13 are given by 13, 26, 39...
The above sequence has a first number equal to 13 and a common difference d = 13.
We need to know the rank of the term 3250.
We use the following formula for the nth term
an = a1 + (n - 1 )d
3250 = a1 + (n - 1 )d
Substitute a1 and d by their values
3250 = 13 + 13(n - 1)
Solve for n to obtain
n = 250
3250 is the 250th term, we can use the following formula to find sum
sn = n (a1 + an) / 2
s250 = 250 (13 + 3250) / 2 = 407875.
Ex 2: Find the sum of the first 55 positive integers divisible by 17.
Sol : The sequence of the first 55 positive integers divisible by 17 is given by 17, 34, 51…
The above sequence has a first term equal to 17 and a common difference d = 4. The nth
term formula to find the 55th term is
an = a1 + (n - 1 )d
a55 = 17 + 17 (55 - 1) = 935
Now we know the first term and last term so we can find the sum of the first 55 terms using the following formula
sn = n (a1+ an) / 2
s55 = 55 (17 + 935) / 2 = 26180.
Arithmetic involves the rules for primary operations that are applicable to whole numbers, integers, rational and irrational numbers, in a single word, to all the real numbers. 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 7, 14, 21, 28, 35… is an arithmetic sequence with common difference d = 5.
Formula for Infinite 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
Example Problems on Infinite Arithmetic Sequence
Ex1: Find the sum of all positive integers, from 13 to 3250 inclusive, which are divisible by 13.
Sol : Sequence of first few numbers of positive integers divisible by 13 are given by 13, 26, 39...
The above sequence has a first number equal to 13 and a common difference d = 13.
We need to know the rank of the term 3250.
We use the following formula for the nth term
an = a1 + (n - 1 )d
3250 = a1 + (n - 1 )d
Substitute a1 and d by their values
3250 = 13 + 13(n - 1)
Solve for n to obtain
n = 250
3250 is the 250th term, we can use the following formula to find sum
sn = n (a1 + an) / 2
s250 = 250 (13 + 3250) / 2 = 407875.
Ex 2: Find the sum of the first 55 positive integers divisible by 17.
Sol : The sequence of the first 55 positive integers divisible by 17 is given by 17, 34, 51…
The above sequence has a first term equal to 17 and a common difference d = 4. The nth
term formula to find the 55th term is
an = a1 + (n - 1 )d
a55 = 17 + 17 (55 - 1) = 935
Now we know the first term and last term so we can find the sum of the first 55 terms using the following formula
sn = n (a1+ an) / 2
s55 = 55 (17 + 935) / 2 = 26180.
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