Important Formulae: Progressions - CSAT Preparation - UPSC

1. Arithmetic Progression Formulae

• a_n = a₁ + (n - 1)d
  
• Number of terms =
  Sum of first n natural numbers
  ⇒ 1 + 2 + 3 … + n =
  Sum of squares of first n natural numbers
  ⇒ 1² + 2² + 3² + … + n² =
  Sum of cubes of first n natural numbers
  ⇒ 1³ + 2³ + 3³ ... + n³ =
• Sum of first n odd numbers
  ⇒ 1 + 3 + 5 … + (2n - 1) = n²
• Sum of first n even numbers
  ⇒ 2 + 4 + 6 ... 2n = n(n - 1)
• If you have to consider 3 terms in an AP, consider {a-d, a, a+d}. If you have to consider 4 terms, consider {a-3d, a-d, a+d, a+3d}
• If all terms of an AP are multiplied with k or divided with k, the resultant series will also be an AP with the common difference dk or d/k respectively.

2. Geometric Progression Formulae

The list of formulas related to GP is given below which will help in solving different types of problems.

• The general form of terms of a GP is a, ar, ar², ar³, and so on. Here, a is the first term and r is the common ratio.
• The nth term of a GP is T_n = ar^n-1
• Common ratio = r = T_n/ T_n-1
• The formula to calculate the sum of the first n terms of a GP is given by:
  S_n = a[(rⁿ – 1)/(r – 1)] if r ≠ 1and r > 1
  S_n = a[(1 – rⁿ)/(1 – r)] if r ≠ 1 and r < 1
• The nth term from the end of the GP with the last term l and common ratio r = l/ [r(n – 1)].
• The sum of infinite, i.e. the sum of a GP with infinite terms is S_∞= a/(1 – r) such that 0 < r < 1.
• If three quantities are in GP, then the middle one is called the geometric mean of the other two terms. 
• If a, b and c are three quantities in GP, then and b is the geometric mean of a and c. This can be written as b² = ac or b =√ac
• Suppose a and r be the first term and common ratio respectively of a finite GP with n terms. Thus, the kth term from the end of the GP will be = ar^n-k.