Sequences, progressions and series

An arithmetic progression has first term $4$ and common difference $7$. Compute the fifteenth term and the sum of the first fifteen terms.

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In an arithmetic progression $a_5=20$ and $a_{12}=48$. Find the common difference, the first term and the sum of the first ten terms.

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Compute the sum of the first $100$ odd numbers, $1+3+5+\cdots +199$.

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A geometric progression has first term $3$ and common ratio $2$. Compute the tenth term and the sum of the first ten terms.

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In a geometric progression $a_3=12$ and $a_6=-96$. Find the common ratio and the first term.

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Show that the sequence $a_n=\frac{3n-1}{n+2}$ is increasing and compute its limit.

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Compute $\underset{n\to +\infty }{\lim }\frac{5n^3-2n}{2n^3+n^2+1}$.

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Compute $\underset{n\to +\infty }{\lim }\left(\sqrt{n+5}-\sqrt{n}\right)$.

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Compute $\underset{n\to +\infty }{\lim }\frac{4^n-1}{2^{2n+1}+3}$.

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Compute the sum of the series $\sum _{k=0}^{\infty }2{\left(\frac{3}{4}\right)}^k$.

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Compute the sum of the series $\sum _{k=2}^{\infty }{\left(\frac{1}{3}\right)}^k$.

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Using the geometric series, write the number $0.\overline{45}=0.454545\dots$ as a fraction.

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Compute the sum of the series $\sum _{k=1}^{\infty }\frac{1}{(k+1)(k+2)}$, writing the general term as $\frac{1}{k+1}-\frac{1}{k+2}$.

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At the end of every year, $300$ euros are deposited for $10$ years into an account at $2\%$ compound interest. Compute the balance immediately after the last deposit. Then compute the present value, at the rate of $2.5\%$, of a perpetuity of $800$ euros a year with the first payment one year from now.

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