Geometric sequences

Introduction to arithmetic sequences

Geometric Progression

Geometric progression example question:

1. write down the first five terms of the geometric progression which has first term 1 and common ratio ½

Answer:

              1^½, ½^½, ¼^½, ⅙^½, ⅛^½, 

2. find the 10th and 20th terms of the GP with first term 3 and common ratio 2.

Answer:

10th

u_n = ar ^n-1

u₁₀ = 3 x (12) ^10-1

u₁₀ = 3 x (12) ⁹

u₁₀ = 3 x 512

u₁₀ = 1536 

20th

u₂₀ = 3 x (2) ^20-1

u₂₀ = 3 x (2) ¹⁹

u₂₀ = 1572.864

3. find the 7th term of the GP 2, -6, 18

Answer:

u_n = ar ^n-1

u₇ = 2 x (-3) ^7-1

u₇ = 2 x (-3) ⁶

u₇ = 1456

Arithmetic progressions

Arithmetic progressions example question: 

1. write down the first five term of the AP with first term 8 and common difference 7 

    

   Answer:

               8th , 15th, 22nd, 29th, 36th, 

2. write down the first five terms of the AP with first term 2 and common difference -5

Answer:

            2th, -3th, -8th, -13th, -18th, 

3, find the 17th term of the AP with first term 5 and common difference 2

Answer:

a = 5

d = 2

u_n = a (n - 1) d

u₁₇ = 5 + (17 - 1) 2

u₁₇ = 5 + (16) 2

u₁₇ = 5 + 32

u₁₇ = 37

Indices

Example :-

1.   Find the values of the unknown in the following equation

a)   10k = 0.01

b)   (1/5^)h = 1

Answer:-

a) 10^k = 0.01

10^k = 1/100

10^k = 1/10^-2

10^k = 10^-2

K = -2

b) (1/5) ^h = 1

(1/5) ^{h =} (1/5) ⁰

H = 0

2.   Use the laws & properties of indices to simplify the given expressions:-

a)   2h^-2 x 4h⁵

b)   RT⁹ ÷ R^-1_T³

Answer:-

a)   2h^-2 x 4h⁵

= 8h-2+5

= 8h³

b)   RT⁹ ÷ R^-1_T³

= R^{1 - - 9} T^9-3

=R²T⁶

3.   Solve the following of indices, evaluate:-

a) 3^x = 27

b) 81 = 9^2x+3

Answer:-

a) 3^x = 27

3^x = 3³

X = 3

   

b) 81 = 9^2x+3

9² = 9^2x+3

2 = 2x + 3

2-3 = 2x

-1 = 2x

-½ = x

Logarithm

Law of Logarithm

Log_a¹ = 0

Log_a^m + log_aⁿ = log_a^{(m x n)}

Log_a^{b n} = n x log_a^b

Log_a^m – log_aⁿ = log_a(^m/n)

Example:-

1.   Write the following indices to logarithms form:-

a)   5² = 25

b)   164²

c)   125 = 5³

Answer:-

a)   Log₅²⁵ = 2

b)   Log₄¹⁶ =2

c)   Log₅¹²⁵ = 3

2.   Write is log₅⁶²⁵?

Log₅⁶²⁵ = ?

5^x = 625

5^x = 54

Log₅⁶²⁵ = 4

3.   Log₇7 + 4log₇^3-2log₇⁹

= 1 + log₇³⁴ – log₇⁹²

= 1 + log₇⁸¹ – log₇⁸¹

= 1

4.   Solve the equation:-

Log₃ (2 + x) = 4

Log₃⁴ = 2 + x

81 = 2 + x

81 – 2 + x

79 = x