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.
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.
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
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:
The sum of the members of a finite arithmetic progression iscalled an arithmetic series.Expressing the arithmetic series in two different ways:Adding both sides of the two equations, all termsinvolving d cancel:Dividing both sides by 2 produces a common form of theequation:
• An alternate form results from re-inserting the substitution: we getfor example, the sum of the terms of the arithmeticprogression given by an = 3 + (n-1)(5) up to the 50th term is
QESTIONS1. A gentleman buys every year Banks cash certificates of value exceeding the last years 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 beRs. a.The difference between the value of the certificates is Rs.300(d = 300).
• 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 in13th year = 1300 + (13 - 1) 300 = Rs.4900
2.The sum of the first 50 terms common to the ArithmeticSequence 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
• 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
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.
• 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-
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 well see that
since all the other terms cancel. If r ≠ 1, we can rearrange theabove 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.
QUESTIONS1. Write down the 8th term in the Geometric Progression 1, 3, 9, ...ANSWER-