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Important Formulas for CAT Progressions

1. Arithmetic Progression Formulae

  • an = a1 + (n - 1)d
    Important Formulas for CAT Progressions
  • Number of terms = Important Formulas for CAT Progressions
    Sum of first n natural numbers
    ⇒ 1 + 2 + 3 … + n = Important Formulas for CAT Progressions
    Sum of squares of first n natural numbers
    ⇒ 12 + 22 + 32 + … + n2 = Important Formulas for CAT Progressions
    Sum of cubes of first n natural numbers
    ⇒ 13 + 23 + 33 ... + n3 = Important Formulas for CAT Progressions
  • 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.

Important Formulas for CAT Progressions


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 Tn = arn-1
  • Common ratio = r = Tn/ Tn-1
  • The formula to calculate the sum of the first n terms of a GP is given by:
    Sn = a[(r– 1)/(r – 1)] if r ≠ 1and r > 1
    Sn = 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)].

Question for Important Formulae: Progressions
Try yourself:
What is the formula to find the sum of the first n terms of an arithmetic progression?
View Solution

  • 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 = arn-k.

Important Formulas for CAT Progressions

The document Important Formulas for CAT Progressions is a part of the UPSC Course CSAT Preparation.
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FAQs on Important Formulas for CAT Progressions

1. What is the formula for finding the nth term of an arithmetic progression?
Ans. The formula for finding the nth term of an arithmetic progression is given by: \(a_n = a_1 + (n-1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
2. How do you find the sum of the first \(n\) terms of an arithmetic progression?
Ans. The sum of the first \(n\) terms of an arithmetic progression can be calculated using the formula: \(S_n = \frac{n}{2}(2a_1 + (n-1)d)\), where \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference.
3. What is the formula for finding the nth term of a geometric progression?
Ans. The formula for finding the nth term of a geometric progression is given by: \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
4. How can you calculate the sum of the first \(n\) terms of a geometric progression?
Ans. The sum of the first \(n\) terms of a geometric progression can be found using the formula: \(S_n = \frac{a_1(1-r^n)}{1-r}\), where \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
5. What is the difference between an arithmetic progression and a geometric progression?
Ans. In an arithmetic progression, each term is obtained by adding a constant difference to the previous term, while in a geometric progression, each term is obtained by multiplying the previous term by a constant ratio.
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