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Arithmetic Progression

An arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
For example, the sequence 3, 5, 7, 9, 11,... is an arithmetic progression with common difference 2.

Arithmetic progression property:

a1 + an = a2 + an-1 = ... = ak+an-k+1

Formulae for the n-th term can be defined as:

an = 1/2(an-1 + an+1)

If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the n-th term of the sequence is given by

an = a1 + (n - 1)d, n = 1, 2, ...

The sum S of the first n values of a finite sequence is given by the formula:

S = 1/2(a1 + an)n, where a1 is the first term and an the last.

or

S = 1/2(2a1 + d(n-1))n

Arithmetic Progression Problems

1) Is the row 1,11,21,31... arithemtic progression?
Solution: Yes it is arithmetic progression with first term 1 and common differnece 10.

2) Find the sum of the first 10 numbers from this arithmetic progression 1, 11, 21, 31...
Solution: we can use this formula S = 1/2(2a1 + d(n-1))n
S = 1/2(2.1 + 10(10-1))10 = 5(2 + 90) = 5.92 = 460

3) Can u proof that if the numbers 1/(c + b) , 1/(c + a), 1/(a + b) are from arithmetic progression then the numbers a2, b2, c2 are also arithmetic progression. If u want to receive the proof write to Dr. Math, please.

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