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Worked solution

Read an arithmetic progression and form its nth term

Write down the first term, common difference and nth term of each AP: (i) −3,−8,−13,−18,−23,…-3,-8,-13,-18,-23,\ldots; (ii) 1,212,4,512,7,…1,2\frac12,4,5\frac12,7,\ldots.

CJC · Sequences and Series · Example 5

(i) Negative common difference

Answer

a=−3a=-3, d=−5d=-5, un=2−5nu_n=2-5n.

Substitute a and d in the nth-term formula

a=−3,d=−8−(−3)=−5,un=−3+(n−1)(−5)=2−5na=-3,\quad d=-8-(-3)=-5,\qquad u_n=-3+(n-1)(-5)=2-5n

The source gives both unsimplified and simplified forms.

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(ii) A fractional common difference

Answer

a=1a=1, d=32d=\frac32, un=32n−12u_n=\frac32n-\frac12.

Keep the common difference exact

a=1,d=52−1=32,un=1+(n−1)32=32n−12a=1,\quad d=\frac52-1=\frac32,\qquad u_n=1+(n-1)\frac32=\frac32n-\frac12

The mixed-number list increases by three halves each time.

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