Revision: Arithmetic Progression Algebra Maths 1 SSC (English Medium) 10th Standard Maharashtra State Board

A sequence is a set of numbers arranged in a definite order.

• Each number in a sequence is called a term.

• A sequence may be written using symbols

  t₁,t₂,t₃,…,t_n
• The general term of a sequence is denoted by t_n.​

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

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

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

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

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

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

In the given AP, let the first be a and the common difference be d.

Then, T_n = a + (n – 1)d

Now, T₄ = a + (4 – 1)d

⇒ a + 3d = 0   ...(1)

⇒ a = –3d 

Again, T₁₁ = a + (11 – 1)d

= a + 10d

= –3d + 10d

= 7d   ...[Using (1)]

Also, T₂₅ = a + (25 – 1)d

= a + 24d

= –3d + 24d

= 21d   ...[Using (1)] 

 i.e., T₂₅ = 3 × 7d = (3 × T₁₁)

Hence, 25^th term is triple its 11^th term.