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Worked solution

Evaluate two geometric terms

Find the sixth and thirteenth terms of a GP with first term a=10a=10 and common ratio r=23r=\tfrac23.

CJC · Sequences and Series · Self-Practice 3

Answer

u6≈1.32u_6\approx1.32, u13≈0.0771u_{13}\approx0.0771 (3 significant figures).

Compare your reasoning

Count multiplications after the first term

u6=10(23)5=320243≈1.32,u13=10(23)12=40960531441≈0.0771.\begin{aligned}u_6&=10\left(\frac23\right)^5=\frac{320}{243}\approx1.32,\\u_{13}&=10\left(\frac23\right)^{12}=\frac{40960}{531441}\approx0.0771.\end{aligned}

The source prints the rounded answers. These exact substitutions are authored working, with powers 5 and 12 rather than 6 and 13.

How did this question go?

Common mistakes

Keep the common ratio exact until the final rounding.