Progressions

7/22/2017www.georgeprep.com learn.georgeprep.com
1
Alosies George
IIM Calcutta
alosies@gmail.com
director@georgeprep.com
8639-71-96-70
Sequences, series and progressions

What are sequences and series?
 A list of things in order
Example
{1, 2, 3, …}
{2, 4, 6, 8}
{a, b, c, d, e, f}
2
Sequence
 Sum of a sequence
Example
1 + 2 + 3 + ….
Series

What are progressions?
 Certain special sequences are called progressions
Arithmetic Progression
Geometric Progression
Harmonic Progression
3
Progressions

Arithmetic Progression
 A sequence a, b, c , … is in arithmetic progression if
𝑏 − 𝑎 = 𝑐 − 𝑏 = common difference (d)
 nth term of an AP = 𝑎 + 𝑛 − 1 𝑑
 Sum of n terms of an AP = 𝑛 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑞𝑒𝑛𝑐𝑒
𝑛
2
(𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 + 𝑙𝑎𝑠𝑡 𝑡𝑒𝑟𝑚)
𝑛
2
[2𝑎 + 𝑛 − 1 𝑑]
4

Properties of arithmetic progression
 AP + k = AP
 AP(k) = AP
 AP1 + AP2 = AP
 AP1 - AP2 = AP
 Corresponding terms and arithmetic mean of an AP
 Choosing terms in an AP
5

Basic Problems - AP
1. If the sum of five terms in an AP is 125, what is the third
term?
2. If the 3rd term and 17th term in an AP are 22 and 148
respectively, what is the 40th term?
3. If 12 times the 12th term of an AP is equal to 18 times
the 18th term of the AP, what is the 30th term of the AP?
4. The ratio of the 4th to the 10th term in an AP is 8 : 22.
What is the ratio of the 15th and 19th term?
6
Answers: Qn 1: 5 Qn 2: 355 Qn 3: 0 Qn 4: 101:129

Geometric Progression
 A sequence a, b, c , … is in Geometric progression if
𝑏
𝑎
=
𝑐
𝑏
= 𝑐𝑜𝑚𝑚𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 (𝑟)
 nth term of an GP = 𝑎𝑟(𝑛−1)
 Sum of n terms of an GP = 𝑎 ∗
(𝑟 𝑛−1)
(𝑟−1)
 Sum of an infinite GP ( only for r <1) =
𝑎
1−𝑟
7

Properties of geometric progression
 GP(k) = GP
 GP1 * GP2 = GP
 Corresponding terms and geometric mean of a GP
 Choosing terms in a GP
8

Basic Problems - GP
If the 10th, 14th and 18th terms of a GP are p, q and r respectively, then
which of the following is true?
1. 2𝑞 = 𝑝 + 𝑟
2. 2𝑞 = 𝑝𝑟
3. 𝑞2 = 𝑝𝑟
4. None of these
9
Answer: Option 3

Harmonic Progression
 A sequence a, b, c , … is in harmonic progression if
1
𝑎
,
1
𝑏
𝑎𝑛𝑑
1
𝑐
𝑎𝑟𝑒 𝑖𝑛 𝐴𝑃
 Harmonic mean between a and b
2𝑎𝑏
𝑎 + 𝑏
10

Relation between AM, GM and HM
11
For positive real numbers
• AM, GM and HM are in GP
• AM ≥ GM ≥ HM

Problem
The sum of n terms of two arithmetic progressions are in the ratio
(7n + 1) : ( 4n + 27). Find the ratio of their 17th terms.
1. 246 : 167
2. 239 : 160
3. Cannot be determined
4. None of these
12
Answer : None of these (232 : 133)

Problems
A ball is dropped from a height of 120 feet and it rebounds
2/3rd of the height. If it continues to fall and rebound, what is
the total distance that the ball would travel before coming to
rest?
1. 720ft
2. 650ft
3. 600ft
4. None of these
13
Answer : 600ft

Problems
What is the value of the expression given below?
1 . 32 + 2 . 52 + 3 . 72 + … … . . 10 𝑡𝑒𝑟𝑚𝑠
1. 14535
2. 19523
3. 13695
4. None of these
14
Answer : 13695

Problems
What is the value of the first n terms of the expression
given below?
5 + 55 + 555 + 5555 + …
1.
5
9
{10 10 𝑛 − 1 }
2.
5
9
{10 10 𝑛 − 𝑛 − 1}
3.
5
9
{10
10 𝑛−1
9
− 𝑛}
4. None of these
15
Ans: option 3

Problems
What is the value of the first n terms of the expression
given below?
1
1 × 2
+
1
2 × 3
+
1
3 × 4
…
1.
n
n+1
2.
n−1
n+1
3.
1
n+1
4. None of these
16
Answer : Option 1

- CAT 2003(L)
Answer : Option 3
17
There are 8436 steel balls, each with a radius of 1
centimetre, stacked in a pile, with 1 ball on top, 3 balls in the
second layer, 6 in the third layer, 10 in the fourth, and so on.
The number of horizontal layers in the pile is
1.34
2.38
3.36
4.32
Problems
Triangular numbers
𝑛 𝑡ℎ
𝑡𝑒𝑟𝑚 =
𝑛 𝑛 + 1
2
Sum of 𝑛 terms =
𝑛 𝑛+1 𝑛+2
6

- CAT 2008
Answer : Option 3
18
The number of common terms in the two sequences 17,
21, 25,..., 417 and 16, 21, 26,..., 466 is
1. 78
2. 19
3. 20
4. 77
5. 22
Problems

- CAT 2008
Answer : Option 1
19
Problems
Find the sum 1 +
1
12 +
1
22 + 1 +
1
22 +
1
32 + ⋯ +
1 +
1
20072 +
1
20082
1. 2008 −
1
2008
2. 2007 −
1
2007
3. 2007 −
1
2008
4. 2008 −
1
2007
5. 2008 −
1
2009

- CAT 2006
Answer : Option 4
20
Problems
A group of 630 children is arranged in rows for a group
photograph session. Each row contains three fewer
children than the row in front of it. Which of the following
number of rows is not possible?
1. 3
2. 4
3. 5
4. 6
5. 7

Progressions

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Progressions

Similar to Progressions (20)

More from George Prep

More from George Prep (20)

Recently uploaded

Recently uploaded (20)

Progressions

Editor's Notes