Chapter 1 Text work review

16. t₁ = 3, r = 3, S_n = 9840, n = ?

9840 = 3(3ⁿ - 1)/(3 - 1) 

9840(\( \frac{2}{3} \)) = 3ⁿ - 1 = 6560

3ⁿ = 6561 = 3⁸

n = 8

17. t₃ = 24, t₄ = 36, S₁₀ = ?

t₄ = rt₃

r = \( \frac{36}{24} \) = \( \frac{3}{2} \)

t₃ = r²t₁

t₁ = 24(\( \frac{2}{3} \))² = 24\( \frac{4}{9} \) = \( \frac{32}{3} \)

S₁₀ = \( \frac{32}{3} \)(\( \frac{( \frac{3}{2})^9 \frac{3}{2} - 1 }{ \frac{3}{2} - 1 } \)) = \( ( \frac{32}{3} )( \frac{ \frac{59049}{1024} - 1 }{ \frac{1}{2} }) = (\frac{64}{3} )( \frac{59049}{1024} - \frac{1024}{1024}) \) = \( ( \frac{64}{3})( \frac{58025}{1024}) \) = \( \frac{58025}{48} \)

18. a + b + c = 35, abc = 1000, what are a, b, c?

b = ar, c = br = ar²

abc = 1000 

a(ar)(ar) = 10³

a³r³ = 10³

ar = 10

r = \( \frac{10}{a} \)

b = ar = a\( \frac{10}{a} \) = 10

a + b + c = 35  

a + 10 + ar² = 35

a + a(\( \frac{100}{a^2} \) = 25

a² - 25a + 100 = 0

(a - 20)(a - 5) = 0

a = 20 or a = 5

a + c = 25, c = 25 - a

c = 5 or c = 20

a = 20, b = 10, c =5   OR   a = 5, b = 10, c = 20

19. S₇ = 89, S₈ = 104, t₈ = ?

t₈ = S₈ - S₇ = 15

20.