Question

What is A.P?

List down Basics formulaes
Class 10 CBSE

Thanks

Answer 1

Hi

Here devil1407

A.P is known as Arithmetic Progression.Calculation found by Young Gauss.

Formulae:
                #General form:
                                     nTH term of A.Pis:
                                                                 tn = a + (n-1)d
                                                                 a = first term
                                                                 d = common difference
                                                                 n= no.of terms
               #Sum of n term: 
                                     Sn = n/2{2a+(n-1)d}
                                               or
                                     Sn = n/2(a+)
                                     where,
                                               l=a+(n-1)d,is the last term.
             
                #The nTH term of an A.P is the difference of the sum of first n terms and the sum to first(n-1) terms of it.i.e:
                                                                     an = Sn - S(n-1)

               # If nTH term of an A.P is odd then, {n+1/2}
               # If nTH term of an A.P is even then,{n/2 + 1}
               # To find d:
                               a2 - a1 = d
                               a3 - a2 = d
                               a4 - a3 = d
                                               ...... so on.

Hope it helps u..
Thanx....Bye

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Answer 2

Hi,

Here is your answer,

Q.1 What is A.P ?

Ans. A sequence in which the difference of two consecutive terms is constant, is called Arithmetic Progression.

Example:- a , a+d , a + 2d is an A.P , where a = first term , d = common difference.

nth Term of A.P:- If a is the first term, d is the common difference and l is the last term of an A.P.

     →       a , a + d , a + 2d , a +3d........... l

(1) nth term is given by l = an = a +(n-1)d

(2) nth term of an A.P from the last term is an = l - (n-1)d

(3) an + a'n = a + l
                  {    nth term from the start + nth term from the end = 1st term + last term. }

(4) Common difference of an A.P.    d = Tn - Tn-1. Here n > 1

(5) Tn = 1/2[Tn-k + Tn+k], k<n

Now, let us learn about properties of A.P

1) If a constant is added or subtracted from each term of an A.P, then the resulting sequence is an A.P with same common difference.

2) If the terms of an AP are chosen at regular intervals, then they form an AP

3) If each term of an A.P is multiplied or divided by a non-zero constant k, then the resulting sequence is also an A.P, with common difference kd or d/k , where d = common difference.

4) If an , an+₁ and an +₂ are three consecutive terms of an A.P, then 2a n + 1 = an + an + 2 

→ Let the three terms of A.P can be taken as:- (a-d) , a , (a+d)

→ Let the four terms of A.P can be taken as:- (a-3d) , (a-d) , (a+d) , (a+3d)

→ Let the five terms of A.P can be taken s:- (a-2d) , (a-d) , a , (a+d), (a+2d)

SUM OF 'N' TERMS OF AN A.P

⊕ Sum of n terms of A.P is given by;-

         →     Sn = n/2[2a+(n-1)d] = n/2 [a+l] where l = last term

⊕ A sequence is an A.P, if the sum of n terms is of the form An² + Bn where A and B are constants and common difference in such cases will be 2A.

⊕ Tn = Sn - Sn-1 

  nth term of  A.P = Sum of n terms - Sum of (n-1) terms

Now, let us jump to our last topic of Arithmetic Mean:-

1) If a , A , b are in A.P then A = a+b/2 is called the Arithmetic Mean of a and b.

2) If a₁ , a₂ , a₃ ........... an  are n, numbers, then their AM is given by,

         ↔       A = a₁ + a₂ + .....+ an / n

3) If a, A₁ , A₂ , A₃ .......... An , b are in A.P then A₁ , A₂ , A₃ .........., An are n arithmetic mean between a and b, whose

            →     d = b - a/n+1

              →   A₁ = a + d = na + b/ n+1

             →   A₂ = a + 2d = (n-1)a + 2b / n + 1 ................. An = a + nd = a +nb/ n+1

 4) Sum of n AM's between a and b is nA

                A₁ + A₂ + A₃ + ........ + An = nA


Hope it helps you !