- A sequence is an ordered list of objects where order matters and elements can repeat. A series is the sum of a sequence's terms. - Sequences whose terms follow patterns are called progressions, including arithmetic and geometric progressions. - An arithmetic progression has a constant difference between consecutive terms, and the nth term can be calculated using a formula involving the first term and common difference. - A geometric progression multiplies the previous term by a fixed ratio, and the nth term can be calculated using a formula involving the first term and common ratio.

2. SEQUENCES
• In mathematics, a sequence is an ordered list of
objects (or events). Like a set, it
contains members (also called elements, or terms).
The number of ordered elements (possibly infinite) is
called the length of the sequence. Unlike a set, order
matters, and exactly the same elements can appear
multiple times at different positions in the sequence. A
sequence is a discrete function.

3. SERIES
• A series is, informally speaking, the sum of the
terms of a sequence. Finite sequences and
series have defined first and last terms,
whereas infinite sequences and series continue
indefinitely.
• The terms of the series are often produced
according to a certain rule, such as by a formula,
or by an algorithm.

4. PROGRESSION
• Those sequence whose terms follow
certain patterns are called progressions.
We can also say that those sequences are called
progressions whose general term can be
determined
• Progression may refer to:
• Arithmetic progression
• Geometric progression

5. ARITHMETIC
PROGRESSION
• In mathematics, an arithmetic progression (AP)
or arithmetic sequence is a sequence of numbers such
that the difference between the consecutive terms is
constant. For instance, the sequence 3, 5, 7, 9, 11, 13,
… is an arithmetic progression with common difference
of 2.
• If the initial term of an arithmetic progression is and the
common difference of successive members is d, then
the nth term of the sequence () is given by:

6. The sum of the members of a finite arithmetic progression is
called an arithmetic series.
Expressing the arithmetic series in two different ways:
Adding both sides of the two equations, all terms
involving d cancel:
Dividing both sides by 2 produces a common form of the
equation:

7. • An alternate form results from re-inserting the
substitution:
we get
for example, the sum of the terms of the arithmetic
progression given by an = 3 + (n-1)(5) up to the 50th term is

8. QESTIONS
1. A gentleman buys every year Bank's cash certificates of
value exceeding the last year's purchase by Rs. 300. After
20 years, he finds that the total value of the certificates
purchased by him is Rs. 83,000. Find the value of the
certificates purchased by him in the 13th year.
ANSWER-
Let the value of the certificates purchased in the first year be
Rs. a.
The difference between the value of the certificates is Rs.300
(d = 300).

9. • Since, it follows Arithmetic progression the total value of
the certificates after 20 years is given by
Sn = = =
By simplifying, we get 2a + 5700 = 8300.
Therefore, a = Rs.1300.
The value of the certificates purchased by him in nth year = a
+ (n - 1) d.
Therefore, the value of the certificates purchased by him in
13th year = 1300 + (13 - 1) 300 = Rs.4900

10. 2.The sum of the first 50 terms common to the Arithmetic
Sequence 15, 19, 23..... and the Arithmetic Sequence 14, 19,
24..... is
• ASWER=The two series are in A.P. Therefore, the
common series will also be in an A.P
Common difference of 1st series = 4 and the common
difference of 2nd series = 5.
Common difference of the sequence whose terms are
common to the two series is given by L.C.M of 4 and 5 =
20

11. • Here the first term of the identical (common terms)
sequence is 19.
• We know the sum of first n terms is given by =
• Hence, the sum of first 50 terms of this sequence =
= 25450

12. GEOMETRIC
PROGRESSION
• In mathematics, a geometric progression, also
known as a geometric sequence, is
a sequence of numbers where each term after the first
is found by multiplying the previous one by a fixed
non-zero number called the common ratio. For
example, the sequence 2, 6, 18, 54, ... is a geometric
progression with common ratio 3. Similarly
10, 5, 2.5, 1.25, ... is a geometric sequence with
common ratio 1/2.

13. • Thus, the general form of a geometric sequence is
• Geometric series-
• The sum of the terms of a geometric progression, or of an
initial segment of a geometric progression, is known as
a geometric series.
• General form of geometric series is-

14. Geometric Series
• A geometric series is the sum of the numbers in a
geometric progression=
• We can find a simpler formula for this sum by multiplying
both sides of the above equation by 1 − r, and we'll see
that

15. since all the other terms cancel. If r ≠ 1, we can rearrange the
above to get the convenient formula for a geometric series:
OR
• General term of geometric progression is given by-
• Where “a” is first term ,r is common ratio and “n” is term.

16. QUESTIONS
1. Write down the 8th term in the Geometric Progression 1,
3, 9, ...
ANSWER-

17. 2.