Arithmetic Progressions — Exercise 5.2

Use aₙ=a+(n−1)d and solve for the missing entry.

(i)

a₈=7+7×3=28.

(ii)

0=−18+9d gives d=2.

(iii)

−5=a+17(−3) gives a=46.

(iv)

3.6=−18.9+(n−1)2.5 gives n−1=9, so n=10.

(v)

With d=0 every term is 3.5; a₁₀₅=3.5.

(i)

For 10,7,4,…, a=10,d=−3. a₃₀=10+29(−3)=−77. Choice C.

(ii)

For −3,−1/2,2,…, d=5/2. a₁₁=−3+10×5/2=22. Choice B.

For equally spaced terms, d=(last−first)/(number of gaps).

(i)

2,□,26 has two gaps: d=(26−2)/2=12. Completed: 2,14,26.

(ii)

□,13,□,3: 13 to 3 spans two gaps, so d=−5. Completed: 18,13,8,3.

(iii)

5,□,□,9½: three gaps give d=(9.5−5)/3=1.5. Completed: 5,6½,8,9½.

(iv)

−4,□,□,□,□,6: five gaps give d=2. Completed: −4,−2,0,2,4,6.

(v)

□,38,□,□,□,−22: four gaps from 38 to −22 give d=−15. Completed: 53,38,23,8,−7,−22.

Here a=3,d=5. Set 78=3+5(n−1). Then 75=5(n−1), so n−1=15.

For 7,13,19,…,205: 205=7+6(n−1), hence n=34. For 18,15½,13,…,−47: −47=18−2.5(n−1), hence n−1=26.

Set −150=11−3(n−1). This gives 3n=164 and n=164/3. A term index must be a positive integer.

Subtract a+10d=38 from a+15d=73: 5d=35, so d=7. Then a=38−70=−32 and a₃₁=−32+30×7.

Subtract a+2d=12 from a+49d=106: 47d=94, so d=2. Thus a=8 and a₂₉=8+28×2.

The difference gives 6d=−12, so d=−2. From a+2d=4, a=8. Set 0=8−2(n−1), giving n−1=4.

a₁₇−a₁₀=(a+16d)−(a+9d)=7d. Thus 7d=7.

Here d=12. The difference aₙ−a₅₄=(n−54)12=132, so n−54=11.

If first terms are a and b, the nth-term difference is [a+(n−1)d]−[b+(n−1)d]=a−b. It is independent of n.

The first is 105 and last 994. Set 994=105+7(n−1): n−1=889/7=127.

They are 12,16,…,248. Thus 248=12+4(n−1), giving n−1=59.

Their nth terms are 63+2(n−1) and 3+7(n−1). Equating gives 60=5(n−1), so n=13. Each term then equals 87.

The term difference is 2d=12, so d=6. Then a+2×6=16 gives a=4.

Reverse the AP: first term 253, difference −5. Its 20th term is 253+19(−5).

The equations are 2a+10d=24 and 2a+14d=44. Subtract to get 4d=20, hence d=5. Then 2a=24−50=−26, so a=−13.

7000=5000+200(n−1) gives n−1=10. The first year is 1995, so add ten calendar years.

20.75=5+1.75(n−1). Thus n−1=15.75/1.75=9.