List questions from geometric-progressions

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How do I solve the following exercise?

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I need some advice on solving this summation

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How do I derive a formula for summation of this series?

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If one Arithmetic Mean (A.M) $A$ and the two Geometric Means (G.M.'s) $G_1$ and $G_2$ are inserted between two given positive numbers, prove that:

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The Arithmetic Mean (A.M) between two numbers exceeds their Geometric Mean (G.M.)

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Help with understand the following inequality steps.

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If $a,b,c$ be in Arithmetic Progression,

Published on 22 Mar 2026 - 22:41

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Induction with a recursive sequence

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convergence of recursive sequence to limit not dependent on first term

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Probability of two consecutive successes before two consecutive failures

Published on 22 Mar 2026 - 14:31

80

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Show that $1+ \frac{1}{p} + \frac{1}{p^2} + ... + \frac{1}{p^k} = \frac{p^{k+1}-1}{p^k(p-1)}$

Published on 23 Mar 2026 - 6:41

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Finding the limit of the recursive sequence defined as $5a_{n+2}=3a_{n+1}+2a_n$

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Summation of binomial coefficients multiplied by their index

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Can we calculate $ i\sqrt { i\sqrt { i\sqrt { \cdots } } }$?

Published on 23 Mar 2026 - 6:55

List Question

i dont know how to solve sum of series $n \cdot \frac{1}{2^n}$

How do I solve the following exercise?

I need some advice on solving this summation

How do I derive a formula for summation of this series?

If one Arithmetic Mean (A.M) $A$ and the two Geometric Means (G.M.'s) $G_1$ and $G_2$ are inserted between two given positive numbers, prove that:

The Arithmetic Mean (A.M) between two numbers exceeds their Geometric Mean (G.M.)

Help with understand the following inequality steps.

If $a,b,c$ be in Arithmetic Progression,

Induction with a recursive sequence

convergence of recursive sequence to limit not dependent on first term

Probability of two consecutive successes before two consecutive failures

Show that $1+ \frac{1}{p} + \frac{1}{p^2} + ... + \frac{1}{p^k} = \frac{p^{k+1}-1}{p^k(p-1)}$

Finding the limit of the recursive sequence defined as $5a_{n+2}=3a_{n+1}+2a_n$

Summation of binomial coefficients multiplied by their index

Can we calculate $ i\sqrt { i\sqrt { i\sqrt { \cdots } } }$?

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