UPSC  >  Important Formulae: Progressions

# Important Formulae: Progressions - CSAT Preparation - UPSC

## 1. Arithmetic Progression Formulae

• an = 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 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)].
• 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.

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## FAQs on Important Formulae: Progressions - CSAT Preparation - UPSC

 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 position of the term, and $$d$$ is the common difference.
 2. How do I find the common difference in an arithmetic progression if only two terms are given?
Ans. To find the common difference in an arithmetic progression when only two terms are given, subtract the first term from the second term. The result will give you the value of the common difference.
 3. Can an arithmetic progression have a negative common difference?
Ans. Yes, an arithmetic progression can have a negative common difference. In an arithmetic progression, the common difference can be positive, negative, or zero. It determines the pattern of the progression.
 4. How can I find the sum of an arithmetic progression?
Ans. The sum of an arithmetic progression can be found using the formula: $S_n = \frac{n}{2}(a_1 + a_n)$ where $$S_n$$ is the sum of the first $$n$$ terms, $$a_1$$ is the first term, and $$a_n$$ is the nth term.
 5. Can an arithmetic progression have a fractional common difference?
Ans. No, an arithmetic progression cannot have a fractional common difference. The common difference in an arithmetic progression must be a whole number. If a fractional common difference is encountered, it suggests that the given sequence is not an arithmetic progression.

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