Progressions concepts for CAT

This is one of the chapters that contributes extremely logical problems that can be readily solved by innovative methods as opposed to conventional methods (such as formula application, etc.). The CAT is a test where logic predominates over formal mathematics.

Therefore, students who use creativity to solve these problems receive higher grades than those who rely on formulas. In the beginning of the chapter, I presented conventional mathematical methods, and then, as the chapter progressed, I introduced novel techniques for quicker and more accurate calculation. Remember that 1–2 questions must be posed on the CAT, and this chapter is also essential for other entrance exams, such as XAT, FMS, IIFT, CET, and MAT.

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Sequence

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A sequence is a collection of numbers occurring in a specific order or according to a rule. Thus, any set of numbers a₁, a₂, a₃,..., a_n such that for each positive integer, there is a number n, i.e., a sequence, is sequential. The numerals a₁, a₂, a₃,..., a_n are referred to as elements (or members) of the sequence.

The number of its elements determines whether a sequence is finite or infinite. If there are a finite number of elements, the sequence is finite; if there are an infinite number of elements, the sequence is infinite.

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Series

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Let u₁,u₂,u₃,...,u_n be provided as a sequence of n integers, where n is a specified positive integer. The sum of these numbers, u₁ + u₂ + u₃ +... + u_n, is referred to as a finite series, while the numbers u₁, u₂, u₃,..., u_n are referred to as terms of the series.

If S_n = u₁ + u₂ + u₃ +... + u_n, then S_n is the sum of n terms (or the sum of the firstn terms) of the series.

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Progressions

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Progression is the name for a certain kind of sequence. When the nth term "a_n" is given, the series "a₁,a₂,...,a_n" is known.

(i) If a_n =2n, then the sequence is 2, 4, 6, 8, ...
(ii) If a_n = 2n −1, then the sequence is 1, 3, 5, 7, ....

Based on the type of sequence , progressions can be of three types.

• Arithmetic Progression – AP
• Geometric Progression – GP
• Harmonic Progression – HP

Arithmetic Progression - AP

A series of numbers a₁, a₂, a₃,..., a_n is said to be in arithmetic progression when a₂ – a₁ = a₃ – a₂ =..., which means that the numbers keep going up or down by the same amount. This constant amount is called the common difference of the AP.

N^th term of an AP is denoted by T_n and is given by

T_n = a + (n-1)d

Last term of an AP is denoted by l and is given by

l = a + (n-1)d

Sum of First n Terms of an A.P.

Let if a be the first term of the progression and d be the common difference, then the

First term T₁= a

Second term T₂ = a+d

Third term T₃ = a+2d

Fourth term T₄ = a+3d

Fifth term T₅ = a+4d and so on.

S_n = T₁+T₂+T₃+T₄+….+T_n

 S_n = a+( a+d) +(a+2d) +(a+3d)+….+a+(n-1)d

 S_n= \(\frac{n}{2}\) [2a + (n-1)d]

Arithmetic mean(AM)

When there are three quantities in AP, the middle one is referred to as the arithmetic mean (A.M.) of the other two.

a, b, c are in AP, then A.M. of a and c i.e. \(b=\frac{a+c}{2}\) = AM (Arithmetic Mean)

How to Insert n Arithmetic Means Between a and b

If a and b are the first and last terms between which we want to insert n arithmetic means m₁, m₂, m₃,..., m_n, then m₁, m₂,..., m_n.with common difference d = \(\frac{b-a}{n+1}\)

Properties of Arithmetic Progressions (AP)

1. When a specific number is added or subtracted from each term of a given AP, the resulting sequence is also an AP and has the same common difference as the given AP.
2. When each term of an AP is multiplied by a constant (or divided by a nonzero constant), the resulting sequence is also an AP.
3. If m₁, m₂, m₃,..., m_k and n₁, n₂, n₃,..., n_k are in AP, then (m₁ +n₁),(m₂ +n₂),(m₃ +n₃),...,(m_k +n_k ) are also in AP.
4. If m₁, m₂, m₃,..., m_k and n₁, n₂, n₃,..., n_k are in AP, then (m₁ –n₁),(m₂ –n₂),(m₃ –n₃),...,(m_k –n_k ) are also in AP
5. If there are an odd number of terms in an AP as a₁, a₂, a₃,..., a_n, then the term in the middle is the A.M.
6. If there are even number of terms in an AP as a₁ , a₂ , a₃ ,..., a_n , then a₁ +a_n = a₂ +a_n–1 = a₃ +a_n–2 = a₄ +a_n–3 etc.
7. The sum of an AP with n terms is given by S_n = n(A.M.)
8. If (m - k), m, and (m + k) are consecutive elements of an AP, and k is the common difference. Similarly, five AP terms can be represented as (m - 2k), (m - k), m, (m + k), and (m + 2k).
9. If we must take 4 AP terms, then the terms are(m - 3k), (m - k), (m + k), (m + 3k), where (m - 3k), (m - k), (m + k), (m + 3k) are consecutive AP terms and (2k) is the common difference. Similarly, six AP terms are as (m–5k), (m–3k), (m–k), (m+k), (m+3k), and (m+5k).
10. The sum of first n natural numbers

\( \Sigma n = 1+ 2+3+4+......+n= \frac{n(n+1)}{2}\)

11. The sum of squares of first ‘n’ natural numbers

\(\Sigma n^2= 1^2+ 2^2+ 3^2+4^2+.......+n^2 = \frac{n(n+1)(2n+1)}{6}\)

12. The sum of cubes of first ‘n’ natural numbers

\(\Sigma n^3= 1^3+ 2^3+ 3^3+ 4^3+........+ n^3= [ \frac{n(n+1)}{6}]^2= [\Sigma n]^2\)

13. The sum of first ‘n’ odd numbers i.e., 1+3 +5 + 7 +... + (2n –1) = n²
14. The sum of first n even numbers i. e. , 2 + 4 + 6 + ... + 2 n = n ( n + 1)

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Geometric Progression-GP

A sequence a₁, a₂, a₃,..., a_n is said to be in Geometric Progression if

 \(\frac{a_2}{a_1}= \frac{a_3}{a_2}= \frac{a_4}{a_3}=.........r\)

And a₁,a₂,a₃,...,a_n are all non-zero numbers and r is said to be the common ratio of the G.P

N^th term of Geometric Progression (GP)

First term T1 = a

Second term T₂ = ar²

Third term T₃ = ar³

Fourth term T₄ = ar⁴

Fifth term T₅ = ar⁵ and so on.

So the n^th term of a GP is given by. T_n = a.r^(n-1) it is also the last term (l) of GP.

S_n=T₁+T₂+T₃+T₄+….+T_n

S_n = \(\frac{a(r^n-1)}{(r-1)}\) , if r>1

S_n = \(\frac{a(1-r^n)}{(1-r)}\) , if r<1 also S_n = \(\frac{lr-a}{r-1}\) , if r>1

Sum of an Infinite G.P.

S_n = \(\frac{a}{1-r}\) ; |r|<1

a is the first term and r is the common ratio of the G.P. of infinite terms.

Geometric Mean (G.M.)

If three quantities are in G.P., the middle one is called the

geometric mean (G.M.) of the other two quantities. If a, m, b are in G.P., then m is the G.M. of a and b.

\(\frac{m}{a}= \frac{b}{m} \)

m² = ab

m = \(\pm \sqrt{ab}\)

if a₁, a₂, a₃,..., a_n are in G.P., then

G.M. = (a₁ .a₂ .a₃ .....a_n )^{\(\frac{1}{n}\)}

Properties of Geometric Progression

1. If each term of a G.P. is multiplied or divided by a certain non-zero constant, then the resulting sequence is also a G.P.
2. If a₁ ,a₂ ,a₃ ,... and b₁ ,b₂ ,b₃ ,... are two geometric progressions, then the sequence a₁ b₁ , a₂ b₂ , a₃ b₃ ,... is also a G.P.
3. If there are odd number of terms in a G.P. as a₁, a₂, a₃,..., a_n then the middle most term is the G.M. of the given sequence.
4. If there are even number of terms in a G.P. as a₁ , a₂ , a₃ ,..., a_n , then

\(\sqrt{a_1.a_n} = \sqrt{a_2.a_{n-2}} = \sqrt{a_3.a_{n-3}}= ......GM\)

5. If we have to take three terms in G.P., it is convenient to take them as (m/k) , m , mk; where (m/k), m and mk are consecutive terms of G.P. and k is the common ratio. Similarly we can take five terms of G.P. as (m/k²),(m/k), m, mk, mk²
6. If a₁, a₂, a₃,..., a_n are in G.P. then a₁^m, a₂^m, a₃^m, a₄^m,……..a_n^m are also in G.P.
7. If a₁,a₂,a₃,...,a_n are in G.P. (for every a_i >0; i∈I⁺ then log a₁, log a₂, log a₃,...are in AP

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Harmonic Progression-HP

A series of quantities a₁, a₂, a₃, a₄ ..., a_n is said to be in H.P. when their reciprocals \(\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3},......., \frac{1}{a_n}\) are in AP

Harmonic Mean (H.M.)

When three quantities are in H.P., the middle one is called the Harmonic Mean (H.M.) between the other two.

If a, H, b are in H.P.

then H is the H.M.of a, b.

 \(\frac{1}{H} - \frac{1}{a} = \frac{1}{b}- \frac{1}{H}\)

 \(\frac{2}{H}= \frac{1}{a} + \frac{1}{b}\)

 \(\frac{2}{H}= \frac{a+b}{ab}\)

 H = \(\frac{2ab}{a+b}\)