Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Arithmetic Progression (A.P.)

If  t_{n+1}-t_n = d (constant), for  n = 1,\ 2,\ 3... , then  (t_n) is called an Arithmetic Sequence or an Arithmetic Progression and the series  \Sigma t_n is called an Arithmetic Series. The constant  d is known as the common difference (c.d) which may be positive or negative.

Partial Sum of an Arithmetic Series -:

If  \Sigma t_n is an Arithmetic series, then  S_n = \dfrac{n}{2} \left[ 2a+(n-1)d \right] which is same as  S_n = \dfrac{n}{2}\ (a+l) , where  a=\text{ first term }\ t_1 ,

Common difference  = d , then  l=\text{ last term }\cdot t_n = a+ (n-1)\ d of the finite series  a+(a+d)+(a+2d)+...+ (a+(n-1)\ d) .

Therefore  \lim_{n \to \infty}\ S_n = \infty  or  -\infty  according as  d > 0 or  d < 0 .

If  d=0,\ S_n=na\to  \infty  or  -\infty  according as  a >0 or  a < 0 .

This shows that an Arithmetic Series always diverges, except the special case when  t_1=0=d .

In this case  t_n=0 for all  n so that sum of the corresponding Arithmetic Series is zero.

Note -:

Sum of the First all  n Counting Number -:

 1+2+...+n=\dfrac{n(n+1)}{2}

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