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CAT 2025 Lesson : Progressions - Concepts & Cheatsheet

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Note: The video for this module contains a summary of all the concepts covered in the Progressions lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

  7. Cheatsheet

111) Where a\bm{ a }a and d\bm{ d }d are the first term and common difference respectively in an AP with n\bm{n}n terms,

(a)nth\bm{n^{th}}nth term = aaa + (nnn – 1)ddd

(b) Average of an AP = Average of First and Last terms = x1+xn2\dfrac {x_1 + x_n}{2}2x1​+xn​​

(c) Sum of terms of an AP = Sn\bm{ S_n} Sn​ = n×Averagen \times Average n×Average

= n×(x1+xn2)=n2×[2a+(n−1)d]n \times \left( \dfrac{x_1 + x_n}{2} \right) = \dfrac {n}{2} \times [2a + (n - 1)d]n×(2x1​+xn​​)=2n​×[2a+(n−1)d]

(d) Number of terms in an AP = n\bm{ n }n

= Last Term−First TermCommon Difference+1\dfrac {Last \ Term - First \ Term}{Common \ Difference} + 1Common DifferenceLast Term−First Term​+1

= xn−x1d+1\dfrac {x_n - x_1}{d} + 1dxn​−x1​​+1

222) Where a\bm{a}a and r\bm{ r} r are the first term and common ratio respectively in a GP with n terms,

(a) nth\bm{ n^{th}}nth term = arn−1ar^{n-1} arn−1

(b) Geometric Mean of GP = GM of First and Last terms =x1×xn\sqrt{x_1 \times x_n}x1​×xn​​

= a×ar(n−1)\sqrt{a \times ar^{(n-1)}}a×ar(n−1)​ = ar(n−1)/2ar^{(n - 1)/2}ar(n−1)/2

(c) Sum of n terms of a GP = Sn\bm{ S_n }Sn​ = a(rn−1)r−1\dfrac{a(r^{n}-1)}{r-1}r−1a(rn−1)​

(d) If a GP has infinite terms and 0<r<10 \lt r \lt 10<r<1, then Sum of infinite terms of the GP = a1−r\dfrac{a}{1 - r}1−ra​

333) A sequence x1,x2,x3,....,xnx_1,x_2,x_3,....,x_nx1​,x2​,x3​,....,xn​ is said to be in Harmonic Progression if 1x1,1x2,1x3,.....,1xn\dfrac{1}{x_1},\dfrac{1}{x_2},\dfrac{1}{x_3},.....,\dfrac{1}{x_n}x1​1​,x2​1​,x3​1​,.....,xn​1​ are in Arithmetic Progression.


(a) If a,b,c a, b, c a,b,c are in HP, then 1b−1a=1c−1b\dfrac{1}{b} - \dfrac{1}{a} = \dfrac{1}{c} - \dfrac{1}{b}b1​−a1​=c1​−b1​ and b=2aca+cb = \dfrac{2ac}{a + c}b=a+c2ac​

(b) In an HP, the middle term is the Harmonic Mean.

444) Sum of first n\bm{ n }n natural numbers = n(n+1)2\dfrac{n(n + 1)}{2}2n(n+1)​

555) Sum of squares of first nnn natural numbers = (n(n+1)(2n+1))6\dfrac{(n(n + 1)(2n + 1))}{6}6(n(n+1)(2n+1))​

666) Sum of cubes of first n\bm{n}n natural numbers = [n(n+1)2]2\left[ \dfrac{n(n + 1)}{2} \right]^2[2n(n+1)​]2

777) Sum of first n \bm{n }n even numbers =n(n+1)n (n + 1)n(n+1)

888) Sum of first n\bm{n }n odd numbers = n2n^2n2
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