A collection of numbers arranged in a definite order
according to a specific rule.
Examples of Sequences:
Even numbers: 2, 4, 6, 8, ...
Odd numbers: 1, 3, 5, 7, ...
Squares: 1, 4, 9, 16, ...
Order Matters!
2. Sequence vs. A.P.
Not all sequences are Arithmetic Progressions. What makes an A.P. special?
Normal Sequence
Rule: Add increasing amounts
1 (+2)➞
3 (+3)➞
6 (+4)➞
10
Difference changes
Arithmetic Progression
Rule: Add a constant number
5 (+3)➞
8 (+3)➞
11 (+3)➞
14
Difference is
CONSTANT
The Common Difference (d)
In an A.P., the constant number added is called the Common Difference.
Formula:d = t2 -
t1 = t3 - t2 ...
'd' can be three things:
Positive (+)
2, 4, 6, 8...
A.P. is increasing
Negative (-)
10, 7, 4, 1...
A.P. is decreasing
Zero (0)
5, 5, 5, 5...
A.P. is constant
The DNA of an A.P.
Master these 4 variables, and you master the chapter.
a
First Term
The starting point. (Also t1)
d
Common Difference
The constant jump between terms.
n
Position
The seat number. (Always a natural number 1, 2, 3...).
tn
Nth Term
The actual value sitting in that seat.
Practice: Identify the A.P.
Q1: Which of the following sequences are A.P.? If it is an A.P., find the next two
terms.
a) 5, 12, 19, 26, ...
b) 2, -2, -6, -10, ...
c) 1, 1, 2, 3, 5, 8, ...
d) 3, 3, 3, 3, ...
Building the Formula
Let's logically build the formula for any term.
Term 1: t1 = a
Term 2: t2 = a + d
Term 3: t3 = a + d + d = a + 2d
Term 4: t4 = a + 3d
Pattern: The number of d's is always 1 less than the position!
The Nth Term Formula
If we want to find any term at position n...
tn =
a +
(n - 1)d
Practice: Find the Nth Term
Q2: Find the 19th term of the following A.P. 7, 13, 19, 25, ...
Practice: Finding 'n'
Q3: How many two-digit numbers are divisible by 4?
Hint: First construct the
A.P. (12, 16, 20 ... 96)
Sum of an A.P. (Gauss's Trick)
How do we add up all the terms quickly? A 10-year-old Carl Friedrich Gauss figured it out when asked to
add numbers from 1 to 100.
1 + 2 + 3 + ... + 98 + 99 + 100 = ?
He realized pairing opposite ends gave the same sum!
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
Since there are 100 numbers, there are 50 pairs.
50 × 101 = 5050
Formulas for Sum of N Terms (Sn)
Formula 1 (First & Last)
Use when you know the First and Last term.
Sn = n/2 × [ t1 + tn ]
Formula 2 (Standard)
Use when you know a, d, and n.
Sn = n/2 × [ 2a + (n-1)d ]
Practice: Sum of an A.P.
Q4: Find the sum of first 123 even natural numbers.
A.P. in Real Life (Word Problem)
Q5: Sachin invested in a National Saving Certificate scheme. In the first year he
invested ₹5000, in the second year ₹7000, in the third year ₹9000 and so on.
Find the
total amount that he invested in 12 years.
Interesting Properties of A.P.
If you perform basic operations on every term of an A.P., the resulting sequence is still an A.P.!
/ Add or Subtract
If you add or subtract the same constant to all terms, it remains an A.P.
The common difference 'd' remains the same.
/ Multiply or Divide
If you multiply or divide all terms by a non-zero constant, it remains an A.P.
The new difference will be (d × constant).
Practice: Direct Problems
Q6: Find the 27th term of the following A.P. 9, 4, -1, -6, -11, ...
Practice: Direct Problems
Q7: In the following A.P., which term is the number -151? 11, 8, 5, 2, ...
Practice: Direct Problems
Q8: Find the 24th term of the A.P. 12, 16, 20, 24, ...
Practice: Direct Problems
Q9: Find the sum of all odd numbers from 1 to 150.
Practice: Direct Problems
Q10: Find the sum of all even numbers between 1 and 350.
Practice: Word Problems
Q11: There is an auditorium with 27 rows of seats. There are 20 seats in the first row,
22 seats in the second row, 24 seats in the third row and so on.
Find the number of seats in the
15th row and also find how many total seats are there in the auditorium.
Practice: Word Problems
Q12: On 1st Jan 2016, Sanika decides to save ₹10, ₹11 on the second day, ₹12 on the
third day and so on.
If she decides to save like this, then on 31st Dec 2016 what would be her
total saving? (Hint: 2016 is a leap year).
Practice: Word Problems
Q13: A man borrows ₹8000 and agrees to repay with a total interest of ₹1360 in 12
monthly instalments. Each instalment being less than the preceding one by ₹40.
Find the amount
of the first and last instalment.
Practice: Word Problems
Q14: Mary got a job with a starting salary of ₹15000 per month. She will get an
increment of ₹100 per month.
What will be her salary after 20 months?
Practice: Word Problems
Q15: Two A.P.s are given: 9, 7, 5, ... and 24, 21, 18, ...
If the
nth term of both the progressions are equal, then find the value of n and the nth
term.