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 + a_n = a₂ + a_n-1 = ... = a_k+a_n-k+1

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

a_n = ¹/₂(a_n-1 + a_n+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

a_n = 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 + a_n)n, where a₁ is the first term and a_n the last.

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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