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The following three terms are in geometric progression: x, 2x + 7, 10x – 7. What is the 6^th term?

A sequence of numbers is defined by the relation $\dfrac{a_{n + 1}}{a_n} = 3^n$ and it is known that $a_1 = 1$. Find the value of $\log_3 a_{100}$.

Read more about A sequence of numbers is defined by the relation $\dfrac{a_{n + 1}}{a_n} = 3^n$ and it is known that $a_1 = 1$. Find the value of $\log_3 a_{100}$.
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Find the sum of

$$ 1 + 2\left( \dfrac{1}{3} \right) + 3\left( \dfrac{1}{3} \right)^2 + 4\left( \dfrac{1}{3} \right)^3 + \ldots + n\left( \dfrac{1}{3} \right)^{n - 1} + \dots $$

MSTE - Mathematics, Surveying and Transportation Engineering
Review Instructor: Jhun Vert
Language: Taglish, more on Tagalog

Get yourself acquainted with the CE board exam problems in MSTE. Experience the level of difficulty you will encounter in the actual board examination and feel the similar time-constraint in solving problems. Learn how to solve the problem in the most efficient way and improve your problem solving strategies from experienced CE review instructor.

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The sum of three numbers is 7 and the sum their reciprocals is 7/5. If these three numbers form into a GP, find their product.

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There are two ways to take this exam, online or offline. Online exam is available to both registered users and to those who preferred not to register. Offline however is available only to registered users as it needs to download the attached .pdf file below. Note that all downloads from this site are available only to verified user accounts. If you wish to take this exam offline, we suggest that you impose to yourself the time limit that we suggest and don't forget to comeback to submit your answer.

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Problem
An equilateral triangle is inscribed within a circle whose diameter is 12 cm. In this triangle a circle is inscribed; and in this circle, another equilateral triangle is inscribed; and so on, indefinitely. Find the sum of the areas of all the triangles.