Arithmetic and Geometric Progression

1. UNIT 3
Progressions
Mr.T.SOMASUNDARAM
ASSISTANT PROFESSOR
DEPARTMENT OF MANAGEMENT
KRISTU JAYANTI COLLEGE,
BANGALORE

2. UNIT 3
PROGRESSIONS
Introduction, Arithmetic Progression, finding
the nth term of AP and sum to nth term of AP,
Insertion of arithmetic means in given terms of
AP and representation of AP, Geometric
Progression, finding nth term of GP, sum to nth
term of GP, Insertion of geometric means in
given geometric progression and also
representation of GP.

3. INTRODUCTION
Sequence Real Numbers:
“A set of real numbers written in
succession according to some rule is said
to form a sequence of real numbers. The
successive numbers in the sequence are
called its terms or elements.”
Sequence {Xn} = X1, X2, X3, ……..,
Xn,…………….
The number’s X1, X2, X3, are called
elements of sequence {Xn} and Xn is
called nth element of sequence {Xn}.

4. Finite Sequence:
“The no. of terms is finite then such
a sequence is called a finite sequence.”
(i.e.) finite sequence has a last element.
(E.g.) {Xn}= 2, 4, 6, 8, ……….32.
Infinite Sequence:
“Infinite sequence is sequence which has
no last term and it is denoted by ∑ Xn.
(E.g.) X1 + X2 + X3+……..+Xn
+……………..

5. Arithmetic Progression
Arithmetic Progression:
Arithmetic Progression is a sequence of real numbers in
which each element of the sequence is obtained by adding (or
subtracting) the same number‘d’ to its previous element.
Definition:
“A sequence of numbers in which different elements
(except first element) are written by increasing or decreasing
its previous elements by the same quantity is called an
Arithmetic Progression (A.P.)”
A.P. = a, a +d, a + 2d, a + 3d, ……………
where ‘a’ is called first element, ‘d’ is fixed no. by which
element are increased or decreased and it is also called as
differences.

6. Arithmetic Progression
(i.e.) 1st element = a
2nd element = a + d
3rd element = a + 2d
…………………….
10th element = a + 9d
nth element of A.P. = a + (n – 1) d and it
is denoted by Tn.
Formula:
Tn = a + (n-1) d

7. Arithmetic Progression
Sum to n terms of an A.P.:
Formula:
Sn = n / 2 [2a + (n – 1) d]
(Or)
Sn = n / 2 [a + l]
where l = a + (n – 1) d

8. Arithmetic Progression
Exercise problems:
1. Find the 20th term of A.P. 2, 6, 10, ……….
2. Which term of A.P. is 5, 13, 21, …… is 181?
3. Find the common difference of A.P. where first
term is 5 and 11th term is 25.
Classwork problems:
1. Find the 6th, 8th and 17th term of A.P. whose nth
term is 4n – 3.
2. If the first term of A.P. is 2, the 20th term is 59, find
32nd term?
3. If 7 times 7th term of A.P. is equal to 11 times its
11th term show that 18th term of A.P.is zero.

9. Arithmetic Progression

10. Arithmetic Progression
Homework problems:
1. Find the 25th term of A.P. 0.3, 1, 1.7, ……….
2. The 10th term of A.P. is 2 and 16th term is – 10. Find
the 11th term.
3. Determine the 2nd term and the rth term of A.P.,
whose 6th term is 12 and 8th term is 22.
4. Find the three numbers which are in A.P. whose
sum is 12 and the sum of their cubes is 408.

11. Sum to nth terms of an A.P.

12. Sum to nth terms of an A.P.

13. Sum to nth terms of an A.P.
Homework problems:
1. Find the sum of the following A.P. 2, 6, 10, …… to
50th term.
2. The sum of n elements of A.P. 25, 22, 19, 16,……
is 116. Find the number of terms and the last term.
3. Find the sum of the first hundred even natural
numbers divisible by 5.

14. Application Problems (A.P.)
Exercise problems:
1. A person buys every year Bank’s cash certificate of
value exceeding the last year’s purchase by 250.
After 20 years, he finds that the total value of the
certificates purchased by him is Rs.72,500. Find
the value of the certificate purchased by him a) in
the first year and b) in the 13th year.
2. A manufacturer of radio sets produced 600 units in
the 3rd year and 700 units in the 7th year. Assuming
the production uniformly increased by a fixed
number every year, find i) production in first year,
ii) total production in 7 years and iii) production in
10th year.

15. Application Problems (A.P.)
Homework problems:
1. A contractor agrees to sink a well 250 feet deep at a
cost of Rs.2.70 for 1st foot, Rs.2.85 for 2nd foot and
an extra 15 paise for additional foot. Find the cost
of the last foot and the total cost.
2. A club consists of members whose ages in A.P., the
common difference being 3 months. If the youngest
member of the club is just 7 years old and the sum
of the ages of all members is 250 years, find the
number of members in the club.

16. Geometric Progression
Geometric Progression:
Definition:
“A Geometric Progression is a sequence of numbers
in which the ratio of every element to its previous element
is a fixed constant. This fixed constant is called common
ratio.”
G.P. elements are a, ar, ar2, ar3, …arn-1, …
where ‘a’ is the first element and r is the common ratio of A.P.
To find common ratio,
r = ar / r (or) r = ar2 / ar (or) ar3 / ar2 =
……. = arn-1 / a r n-2 …….. = r
where arn-1 is called the nth element or general terms and
it is denoted by Tn.

17. Geometric Progression
Sum to n terms of an A.P.:
Formula:
Sn = a (1 – rn) / (1 - r)
where ‘a’ is first element and ‘r’is the
common ratio.

18. Geometric Progression (G.P.)

19. Geometric Progression (G.P.)

20. Geometric Progression (G.P.)
Exercise problems:
1. The ratio of 9th element of a G.P. to the 6th element
is – 8 and the 5th element is 16. Find the G.P.
2. The 10th element of a G.P. is double the 12th
element. If the 3rd element is 6, find the 5th element.
3. Which element of G.P. 1, -2, 4, -8, …….. Is 1024.
4. The three numbers whose sum is 18 are in A.P. If 2,
4 and 11 are added to them respectively, the
resulting numbers are in G.P. Find the numbers.

21. Geometric Progression (G.P.)
Classwork problems:
1. The second, third and sixth elements of an A.P. are
consecutive terms of a G.P. Find the common ratio
of the G.P.?
2. Find the three numbers which are in G.P., if their
sum is 28 and their product is 512.

22. Geometric Progression (G.P.)

23. Sum to nth terms of an G.P.

24. Sum to nth terms of an G.P.
Classwork problems:
1. The sum of first ten elements of a G.P. is equal to
244 times the sum of five elements. Find the
common ratio.
2. The first and last elements of a G.P. are 3 and 768
respectively and the sum is 1533. Find the common
ratio and the number of terms.
3. Find the sum to n terms of 7 + 77 + 777 + 7777 +
…..

25. Sum to nth terms of an G.P.
Homework problems:
1. Find the sum of the G.P. 27, 9, 3, 1, …. to 8
element.
2. How many elements of the G.P. 1, 2, 4, 8, ………
must be taken amount to 255?
3. Find the sum to n terms of 1 + 11 + 111 + 1111 +
…...

26. Insertion of AM and GM

27. Arithmetic & Geometric means

28. End of the Unit 3
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