Exercise 25

The tenth term of a 10-term geometric progression is 10, and the ratio is 0.1. Then the third term is:

  1. \( 10^8 \)

  2. \( 10^9 \)

  3. \( 10^{10} \)

  4. \( 10^{11} \)

  5. \( 10^3 \)

Try to solve it before checking the answer.

  1. \( 10^8 \)

From Geometric Progressions we have that:

\[ a_n = a_1 r^{n-1} \Rightarrow a_1 = \frac{a_n}{r^{n – 1}} \]

In this case, \( r = 0.1 \),   \( n = 10 \)   and   \( a_{10} = 10 \). Now, we have:

\[ \begin{aligned} a_1 &= \frac{10}{ (0.1)^9 } \\[1em] &= \frac{ 10 }{ \left( \frac{1}{10} \right)^9 } \\[1em] &= \frac{10}{ \frac{1}{10^9} } \\[1em] &= 10 \times 10^9 \\[1em] &= 10^{10} \end{aligned} \]

Hence,

\[ \begin{aligned} a_3 &= a_1 r^{3 – 1} \\[.5em] &= 10^{10} (0.1)^2 \\[.5em] &= 10^{10} \left( \frac{1}{10} \right)^2 \\[.5em] &= 10^{10} \left( \frac{1}{10^2} \right) \\[.5em] &= 10^{10 – 2} \\[.5em] &= \boldsymbol{10^8} \end{aligned} \]

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