Tenth - Mathematics - Arithmetic Sequences

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Here some examples of Sequences:

No.	Sequences	Defference	First Term	5th term
1	5,10,15,20,25,...	5,5,5,5,5,...	5	25
2	1,4,9,16,25,...	3,5,7,9,...	1	25

      A set of numbers written like 1,2,3,4,... or 1,4,9,16,... as the first, second, third and so on, according to a particular rule is called Number Sequences.

you can find varieties of number sequences in your surroundings, ie, Roll numbers of students, Time, ,etc.

      The numbers forming a sequence are called terms.

Example: In the number sequence 5,10,15,20,... 5 is first term, 10 is second term, 15 is also third term and so on.

A sequence got by starting with any number and adding a fixed number repeatedly is called an Arithmetic Sequence.
In other words, An Arithmetic Sequence is a sequence in which we get the same number on subtracting from any term, the term immediately preceding it.

the constant difference got by subtracting from any term the just previous term, is called Common Difference of an arithmetic sequence.

Equations

Taking the first term of an arithmetic sequence as f and the common difference as d, the nth term(nt) is:
     nt=f+(n-1)d
     dn+(f-d)

Taking the first term of an arithmetic sequence as f and the common difference as d and the last term as l, the total number of terms(tt) is:
     tt=(l-n)/d+1

If a,b and c are continues terms of an arithmetic sequence, algibra to find b is:
     b=(a+c)/₂

The sum(s) of all terms of an arithmetic sequence, when n is equal to number of terms, f is equal to first term, l is equal to last term and d is equal to common difference, is:
     s=n/₂(a+n*d)
     s=n/₂(f+l)

Sum(s) of first n even number is:
     s=n(n+1)

Sum(s) of first n odd numbers is:
     s=n²