Sequence and Series
1.0 Introduction
1.0 Introduction
Sequence: Arrangement of numbers following/satisfying a particular rule in which its domain is a set of natural numbers.
For example:
$\begin{equation} \begin{aligned} 2,4,6,8....; \\ 1,4,9,16...; \\ 1,3,9,27...; \\\end{aligned} \end{equation} $
are each a sequence.
Sequence is said to be finite or infinite sequence according it has finite or infinite number of terms.
Series: We get a series by adding or subtracting the terms of a sequence.
For example:
$\begin{equation} \begin{aligned} 2 + 4 + 6 + 8 + ...; \\ 1 + 4 + 9 + 16 + ...; \\ 1 - 3 + 9 - 27 + ...; \\\end{aligned} \end{equation} $
are each a series.
Question 1. Write the sequence whose $n$th term is $\frac{1}{{{n^2}}}\sin \left( {\frac{{n\pi }}{3}} \right)$.
Solution: Let $${a_n} = \frac{1}{{{n^2}}}\sin \left( {\frac{{n\pi }}{3}} \right)$$
Putting $n = 1,2,3,4,...$, we get,
$$\begin{equation} \begin{aligned} {a_1} = \frac{1}{{{1^2}}}\sin \left( {\frac{\pi }{3}} \right) = \frac{{\sqrt 3 }}{2} \\ {a_2} = \frac{1}{{{2^2}}}\sin \left( {\frac{{2\pi }}{3}} \right) = \frac{{\sqrt 3 }}{8} \\ {a_3} = \frac{1}{{{3^2}}}\sin \left( {\frac{{3\pi }}{3}} \right) = 0 \\ {a_4} = \frac{1}{{{4^2}}}\sin \left( {\frac{{4\pi }}{3}} \right) = - \frac{{\sqrt 3 }}{{32}} \\\end{aligned} \end{equation} $$
Hence, the sequence is $$\frac{{\sqrt 3 }}{2},\frac{{\sqrt 3 }}{8},0, - \frac{{\sqrt 3 }}{{32}},...$$
Question 2. A sequence of numbers ${a_1},{a_2},{a_3},...$ satisfy the relation ${a_{n + 1}} = {a_n} + {a_{n - 1}}$ for $n \geqslant 2$. Find ${a_4}$, given ${a_1} = {a_2} = 1$.
Solution: Putting $n=2$ in the given relation, we get $${a_3} = {a_2} + {a_1} = 1 + 1 = 2$$ Again put $n=3$, we get $${a_4} = {a_3} + {a_2} = 2 + 1 = 3$$
