1.2 Geometric Sequences

Charlie and Daniella each began a fitness programme. On day one, they both ran 500 m. On each subsequent day, Charlie ran 100 m more than the previous day whereas Daniella increased her distance by 2 % of the distance ran on the previous day.

(a) Calculate how far

(i) Charlie ran on day 20 of his fitness programme.

(ii) Daniella ran on day 20 of her fitness programme.

On day n of the fitness programmes Daniella runs more than Charlie for the first time.

(b) Find the value of n . 

(a)

(i)

attempt to find  using an arithmetic sequence
e.g.  u₁ = 500 and d = 100

(Charlie ran) 2400 m

(ii)

r = 1.02

attempt to find u₂₀ using a geometric sequence
e.g. identifying u₁ = 500  and a value for   

(Daniella ran) 728 m (728.405...)

(b) `500 * 1.02^(n-1) >= 500 + (n - 1) * 100`

attempt to solve inequality

n > 184.215

n = 185

In the first month of a reforestation program, the town of Neerim plants 85 trees. Each subsequent month the number of trees planted will increase by an additional 30 trees.

The number of trees to be planted in each of the first three months are shown in the following table.

(a) Find the number of trees to be planted in the 15th month.

(b) Find the total number of trees to be planted in the first 15 months.

(a) use of the n^th term of an arithmetic sequence formula

u₁₅= 85 + (15 - 1) x 30 = 805

(b) use of the sum of  terms of an arithmetic sequence formula

`S_15 = 15/2 (85 + 505)` = 4430 (4425)

(c) `4425/15`

295

On 1 January 2025, the Faber Car Company will release a new car to global markets. The company expects to sell 40 cars in January 2025. The number of cars sold each month can be modelled by a geometric sequence where r = 1.1

(a) Use this model to find the number of cars that will be sold in December 2025.

(b) Use this model to find the total number of cars that will be sold in the year

(i)  2025

(ii) 2026

(a)

at least three correct terms of the sequence

`u_12 = 40 xx 1.1^(12 - 1)`

114 (114.124)

(b)

(i)

a sum of at least the first three terms

`S_12 = (40(1.1^12 - 1)) / (1.1 - 1); S_12 = 855 (855.371...)`

(ii) 

`S_24 = 3539.89...; u_13 = 125 or 126.`

The sum of the first  terms of a geometric sequence is given by

`S_n = sum_(r=1)^n 2/3 (7/8)^r`

(a) Find the first term of the sequence, u₁

(b) Find `S_oo`

(c) Find the least value of  such that `S_oo - S_n < 0.001`

(a) attempt to substitute into geometric sequence formula for twelfth term

(b)

attempt to substitute into the geometric series formula
a sum of at least the first three terms

`S_12 = (40 (1.1^12 - 1)) / (1.1 - 1), S_12 = 855 (855.371...)`

finding

`S_24 = 3539.89...; sum_(13)^24 (40 xx 1.1^(n-1)) = 2680 (2684.52..., 2685)`

(c)
attempt to substitute their values into the inequality or formula for S_n

`14/3 - sum_(r=1)^n 2/3 (7/8)^r < 0.001`

attempt to solve their inequality using a table, graph or logarithms (must be exponential)

correct critical value or at least one correct crossover value
`S_oo - S_63 = 0.001036...`

least value is n = 64

The sum to infinity of a geometric series is 32. The sum of the first four terms is 30 and all the terms are positive.

Find the difference between the sum to infinity and the sum of the first eight terms.

`S_oo = a / (1 - r) = 32 and S_4 = (a(1 - r^4)) / (1 - r) = 30`

`a = 32 (1 - r)`

`32 (1 - r^4) = 30`

`r = 0.5 and a = 16`

`S_oo - S_8 = 32 - 16 ((1 - 0.5^8) / (1 - 0.5)) = 0.125 = 1/8`

(a) The sum of the first six terms of an arithmetic series is 81. The sum of its first eleven terms is 231. Find the first term and the common difference.

(b) The sum of the first two terms of a geometric series is 1 and the sum of its first four terms is 5. If all of its terms are positive, find the first term and the common ratio.

(c) The r^th term of a new series is defined as the product of the r^th term of the arithmetic series and the r^th term of the geometric series above. Show that the r^th term of this new series is `(r + 1) 2^(r-1)` .

(d) Using mathematical induction, prove that

`sum_(r=1)^n (r + 1) 2^(r-1) = n 2^n, n in ZZ^+`

(a)

`S_6 = 81 => 81 = 6/2 (2a + 5d)`

`=> 27 = 2a + 5d`

`S_11 = 231 => 231 = 11/2 (2a + 10d)`

`=> 21 = a + 5d`

solving simultaneously, a = 6, d = 3

(b)

`a + ar = 1 a + ar + ar^2 + ar^3 = 5`

`=> (a + ar) + ar^2(1+r) = 5 => 1 + ar^2 xx 1/a = 5`

`r^2 - 4 = 0 => r = +-2`

r = 2

(since all terms are positive)

a = `1/3`

(c)

AP r^th term is 3r +3 

GP r^th term is

`1/3 * 2^(r-1) 3(r + 1) xx 1/3 2^(r-1) = (r + 1) 2^(r-1)`

(d)

prove: 

`: P_n: sum_(r=1)^n (r + 1) 2^(r-1) = n 2^n, n in ZZ^+`

show true for n = 1 , i.e.

LHS = 2 x 2⁰ = 2 = RHS

assume true for n = k , i.e.

`sum_(r=1)^k (r + 1) 2^(r-1) = k 2^k, k in ZZ^+`

consider n = k + 1

`sum_(r=1)^(k+1) (r + 1) 2^(r-1) = k 2^k + (k + 2) 2^k = 2^k (k + k + 2) = 2 (k + 1) 2^k = (k + 1) 2^(k+1)`

hence true for  n = k + 1 is true whenever P_k is true, and P₁ is true, therefore P_n  is true for `n in ZZ^+`

Consider a geometric sequence with first term  and common ratio .  is the sum of the first  terms of the sequence.

(a) Find an expression for S_n in the form `n (a^n - 1) / b`, where `a, b in ZZ^+`

(b) Hence, show that `S_1 + S_2 + S_3 + ... + S_n = (10 (10^n - 1) - 9n) / 81`

(a)

`S_n = (10^n - 1) / 9 (a = 10, b = 9)`

(b)

`S_1 + S_2 + S_3 + ... + S_n = (10 - 1) / 9 + (10^2 - 1) / 9 + ... + (10^n - 1) / 9 = (10 - 1 + 10^2 - 1 + 10^3 - 1 + ... + 10^n - 1) / 9`

attempt to use geometric series formula on powers of 10, and collect -1’s together

`10 + 10^2 + 10^3 + ... + 10^n = (10 (10^n - 1)) / (10 - 1) and - 1 - 1 - 1... = -n = (10 (10^n - 1)) / (10 - 1) * 1/9 - n = (10 (10^n - 1) - 9n) / 81`

Consider the infinite geometric sequence `3, 3(0.9), 3(0.9)^2, 3(0.9)^3, ....`

(a) Write down the 10^th term of the sequence. Do not simplify your answer.

(b) Find the sum of the infinite sequence.

(a) 

`u_10 = 3 (0.9)^9`

(b) recognizing r = 0.9
correct substitution

`S = 3 / (1 - 0.9); S = 3 / 0.; S = 30`

The first three terms of an infinite geometric sequence are 32, 16 and 8.

(a) Write down the value of r .

(b) Find u₆

(c) Find the sum to infinity of this sequence.

(a)

`r = 16/32 (= 1/2)`

(b)

correct calculation or listing terms 

`e.g. 32 xx (1/2)^(6-1) , 8 xx (1/2)^3 , 32,...4,2,1; u_6 = 1`

(c) 

evidence of correct substitution in `S_oo`

`e.g. 32 / (1 - 1/2) , 32 / (1/2); S_oo = 64`

The geometric sequence `u_1, u_2, u_3, ...` has common ratio r . Consider the sequence `A = {a_n = log_2 |u_n| : n in ZZ^+}`

(a) Show that A is an arithmetic sequence, stating its common difference d in terms of r. A particular geometric sequence has u₁ = 3 and a sum to infinity of 4.

(b) Find the value of d.

(a)

state that `u_n = u_1 r^(n-1)` (or equivalent)

attempt to consider a_n and use of at least one log rule

`log_2 |u_n| = log_2 |u_1| + (n - 1) log_2 |r|`

(which is an AP) with d = `= log_2 |r| (and 1^(st) term log_2 |u_1|)`

so  is an arithmetic sequence

(b)

attempting to solve

`3 / (1 - r) = 4; r = 1/4; d = -2`