Sequences and Series
Sequences and Series PDF Notes, Important Questions And Synopsis
Sequence and Series PDF Notes, Important Questions and Synopsis
SYNOPSIS
- A sequence is an ordered list of numbers and has the same meaning as in conversational English. A sequence is denoted by <an>(n ≥ 1) = a1,a2,a3, … an.
- The various numbers occurring in a sequence are called its terms.
- A sequence containing finite number of terms is called a finite sequence. A finite sequence has a last term.
- A sequence which is not a finite sequence, i.e. containing infinite number of terms is called an infinite sequence. There is no last term in an infinite sequence.
- A sequence is said to be an arithmetic progression if every term differs from the preceding term by a constant number. For example, the sequence a1, a2, a3,… an is called an arithmetic sequence or an AP if an+1 = an + d, for all n Î N, where ‘d’ is a constant called the common difference of the AP.
- ‘A’ is the arithmetic mean of two numbers ‘a’ and ‘b’ if form an arithmetic progression.
- A sequence is said to be a geometric progression or GP if the ratio of any of its terms to its preceding term is the same throughout. A constant ratio is the common ratio denoted by ‘r’.
- If three numbers are in GP, then the middle term is called the geometric mean of the other two.
- Some Concepts
- A sequence has a definite first member, second member, third member and so on.
- The nth term <an> is called the general term of the sequence.
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Fibonacci sequence 1, 1, 2, 3, 5, 8,… is generated by the recurrence relation given by
a1 = a2 = 1
a3 = a1 + a2…
an = an-2 + an-1, n > 2 -
If the number of terms are three with common difference 'd', then the three terms can be taken as a – d, a, a + d.
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If the number of terms are four with common difference '2d', then the terms can be taken as a – 3d, a – d, a + d, a + 3d.
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If the number of terms are five with common difference 'd', then the terms can be taken as a – 2d, a – d, a, a + d, a + 2d.
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If the number of terms are six with common difference '2d', then the terms can be taken as a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d.
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General form of an AP is a, a + d, a + 2d,… a + (n - 1)d, where ‘a’ is called the first term of the AP and ‘d’ is called the common difference of the AP. ‘d’ can be any real number.
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If d > 0, then the AP is increasing. If d < 0, then the AP is decreasing. If d = 0, then the AP is constant.
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If ‘a’ is the first term and ‘d’ is the common difference of an AP with 'm' terms, then the nth term from the end is the
term from the beginning.
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General term of a GP is arn-1, where ‘a’ is the first term and r is the common ratio.
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If the number of terms of a GP is 3 with the common ratio r, then the selection of terms can be
.
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If the number of terms of a GP is 4 with the common ratio r2 , then the selection of terms can be
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If the number of terms of a GP is 5 with the common ratio r, then the selection of terms can be
- If a constant is added to each term of an AP, then the resulting sequence is also an AP.
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If a constant is subtracted from each term of an AP, then the resulting sequence is
also an AP. -
If each term of an AP is multiplied by a constant, then the resulting sequence is also an AP.
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If each term of an AP is divided by a non-zero constant, then the resulting sequence is also an AP.
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The arithmetic mean A of any two numbers ‘a’ and ‘b’ is given by
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General form of a GP is a, ar, ar2, ar3…, where ‘a’ is the first term and ‘r’ is the constant ratio which can be any non-zero real number.
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A sequence in geometric progression will remain in geometric progression if each of its terms is multiplied by a non-zero constant.
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A sequence obtained by multiplying two GPs term by term will result in a GP with a common ratio as the product of the common ratios of the two GPs.
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Reciprocals of the terms of a given GP form a GP with the common ratio
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If each term of a GP is raised to the same power, then the resulting sequence also forms a GP.
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The geometric mean (GM) of any two positive numbers ‘a’ and ‘b’ is given by
Some Special Series
- Sum of the first ‘n’ natural numbers:
- Sum of the squares of the first n natural numbers:
- Sum of the cubes of the first n natural numbers:
- Sum of the powers of 4 of the first n natural numbers:
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