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Worked solution

Form the nth term of a geometric progression

Write down the first term, common ratio and nth term of each GP: (i) −1,2,−4,8,−16,…-1,2,-4,8,-16,\ldots; (ii) 1,1/2,1/4,1/8,1/16,…1,1/2,1/4,1/8,1/16,\ldots.

CJC · Sequences and Series · Example 9

(i) Alternating and growing

Answer

a=−1,r=−2a=-1,r=-2, un=(−1)(−2)n−1=12(−2)nu_n=(-1)(-2)^{n-1}=\tfrac12(-2)^n.

Use n minus one multiplications

a=−1,r=2−1=−2,un=(−1)(−2)n−1=12(−2)na=-1,\qquad r=\frac2{-1}=-2,\qquad u_n=(-1)(-2)^{n-1}=\frac12(-2)^n

Both equivalent forms are printed in the Teacher table.

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(ii) Halving each time

Answer

a=1,r=12a=1,r=\tfrac12, un=1(12)n−1=(12)n−1u_n=1(\tfrac12)^{n-1}=(\tfrac12)^{n-1}.

Keep the first term separate from the ratio

a=1,r=12,un=1(12)n−1=(12)n−1a=1,\quad r=\frac12,\qquad u_n=1\left(\frac12\right)^{n-1}=\left(\frac12\right)^{n-1}

There are n−1 multiplications after the initial 1.

How did this part go?