Formulas of Sequence and Series
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The $$nth$$ term $${a_n}$$ of the Arithmetic Progression (A.P) $$a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots $$ is given by $${a_n} = a + (n – 1)d$$.
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The arithmetic mean between $$a$$ and $$b$$ is given by $$A.M = \frac{{a + b}}{2}$$.
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If $${S_n}$$ denotes the sum up to $$n$$ terms of A.P. $$a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots $$ then $${S_n} = \frac{n}{2}(a + l)$$ where $$l$$ stands for the last term, $${S_n} = \frac{n}{2}[2a + (n – 1)d]$$
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The sum of $$n$$ A.M’s between $$a$$ and $$b$$ is $$ = \frac{{n(a + b)}}{2}$$.
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The $$nth$$ term $${a_n}$$ of the geometric progression $$a,{\text{ }}ar,{\text{ }}a{r^2},{\text{ }}a{r^3}, \ldots $$ is $${a_n} = a{r^{n – 1}}$$.
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The geometric mean between $$a$$ and $$b$$ is $$ G.M = \pm \sqrt {ab} $$.
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If $${S_n}$$ denotes the sum up to $$n$$ terms of G.P is $${S_n} = \frac{{a(1 – {r^n})}}{{1 – r}};{\text{ }}r \ne 1$$, $${S_n} = \frac{{a – rl}}{{1 – r}};{\text{ }}l = a{r^n}$$ where $$\left| r \right| < 1$$
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The sum $$S$$ of infinite geometric series is $$S = \frac{a}{{1 – r}};{\text{ }}\left| r \right| < 1$$
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The $$nth$$ term $${a_n}$$ of the harmonic progression is $${a_n} = \frac{1}{{a + (n – 1)d}}$$.
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The harmonic mean between $$a$$ and $$b$$ is $$H.M = \frac{{2ab}}{{a + b}}$$.
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$${G^2} = A \cdot H$$ and $$A > G > H$$; where $$A,G,H$$ are usual notations.
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