Revision: Algebra >> Arithmetic Progression Maths (English Medium) ICSE Class 10 CISCE

When the numbers (terms) in a sequence are connected to each other by a positive (plus) sign or a negative (minus) sign, the sequence becomes a series. 

A progression is a sequence where each term follows a uniform rule.

• Every progression is a sequence, but with a clear pattern.

A sequence is a group of numbers arranged in a definite order following a rule.

• Numbers in a sequence are called terms or elements.

• The term at position n is called the nth term, denoted by Tₙ.

An Arithmetic Progression (A.P.) is a sequence in which the difference between consecutive terms is constant.

• Common difference = d = second term − first term

• The general form of an AP is a, a + d, a + 2d, a + 3d, …
  a = first term
  d = common difference

t_n​ = a + (n − 1)d

Used when the first term a, common difference d, and term number n are known

l = a + (n − 1)d

n^th term from end = l − (n − 1) d

Can be used only after finding:

• last term l

• number of terms

Common difference:

d = t₂ − t₁​

Nature of A.P.

• d > 0→ Increasing

• d < 0→ Decreasing

• $=0$ → Constant

\[2+4+\cdots+2n=n(n+1)\]

\[1+3+\cdots+(2n-1)=n^2\]

If a, n, l are known → \[S_n=\frac{n}{2}(a+l)\]

If a, n, d are known → \[S_n=\frac{n}{2}\left[2a+(n-1)d\right]\]

\[1+2+\cdots+n=\frac{n(n+1)}{2}\]

Arithmetic Mean between a and b = \[\frac{a+b}{2}\]

A.M. is nothing but the middle term of an A.P.

• Three terms in A.P.

  a − d,  a,  a + d​

• Four terms in A.P.

  a − 3d,  a − d,  a + d,  a + 3d​

• Five terms in A.P.

  a − 2d,  a − d,  a,  a + d,  a + 2d​

• Six terms in A.P.

  a − 5d,  a − 3d,  a − d,  a + d,  a + 3d,  a + 5d

• Same number added/subtracted to each term → A.P.

• Same non-zero number multiplied/divided → A.P.

• If a, b, c are in A.P., then
  $(b+c),(c+a),(a+b)$ are also in A.P.