“Respect Is The Key Determinant Of High-Performance Leadership. How Much People Respect You Determines How Well They Perform.”
By a sequence, we mean an arrangement of number in definite order according to some rule. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3, ….k}. A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence.
Let a_{1}, a_{2}, a_{3}, … be the sequence, then the sum expressed as a_{1} + a_{2} + a_{3} + … is called series. A series is called finite series if it has got finite number of terms.
An arithmetic progression (A.P.) is a sequence in which terms increase or decrease regularly by the same constant. This constant is called common difference of the A.P. Usually, we denote the first term of A.P. by a, the common difference by d and the last term by l. The general term or the nth term of the A.P. is given by
a_{n}= a + (n – 1) d.
The sum S_{n }of the first n terms of an A.P. is given by
\( S_n= \frac{n}{2}(2a + (n – 1) d) \)
The arithmetic mean A of any two numbers a and b is given by \( \frac{a + b}{2}\) i.e., the sequence a, A, b is in A.P.
A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by a and its common ratio by r. The general or the nth term of G.P. is given by a_{n}= ar^{n – 1} The sum S_{n} of the first n terms of G.P. is given by \( S_n= \frac{a(r^n – 1)}{(r – 1)} = \frac{a(1 – r^n)}{(1 – r)}, {\rm if } r \ne 1 \).
The geometric mean (G.M.) of any two positive numbers a and b is given by \( \sqrt{ab}\) i.e., the sequence a, G, b is G.P.
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