Question

The last 3 terms of the sequence: 4; x; y; 18 form a geometric progression whilst the first 3 terms form an arithmetic progression. 1.2.1 Calculate the values of x and y if both are natural numbers. 1.2.2. Hence, calculate T15 of the geometric sequence. (Correct your answer to ONE decimal place)​

Answer 1

In the sequence, the value of y is 12 while the value of x is 8.

Equation

An equation is used to show the relationship between two or more variables.

The last three terms x, y, 18 form a geometric progression, hence:

r (common ratio) = y/x = 18/y

y²/18 = x    (1)

Also, the first 3 terms 4, x, y form an arithmetic progression, hence:

common difference (d) = x - 4 = y - x

x - 4 = y - x

2x - y - 4 = 0

2(y²/18) - y - 4 = 0

y = 12

x = (12)²/18 = 8

In the sequence, the value of y is 12 while the value of x is 8.

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Answer 2

The value of x and y are 8 and 12 respectively.

The value of T₁₅ of the geometric progression is 1167.7

The first 3 terms forms arithmetic progression.

Using Arithmetic progression formula

AP formula

• aₙ = a + (n - 1)d

where

a = first term

d = common difference

n= number of term

let's find the third term

y = 4 +(3 - 1)(x-4)

y = 4 + 2(x - 4)

y = 4 + 2x - 8

y - 2x = -4

y = 2x - 4

The last three terms are geometric progression. Therefore,

• aₙ = arⁿ⁻¹

where

a = first term

r = common ratio

Therefore,

a = x

r = 18 / y

18 = x(18 / y)²

18 = 324x / y²

18y² = 324x

x = 18y² / 324

x = y² / 18

let's combine the equations

y = 2(y² / 18) - 4

y = y² / 9 - 4

y - y² / 9 = -4

9y - y²/9 = -4

-y² + 9y = -36

y² - 9y - 36 = 0

y² + 3y - 12y - 36 = 0

(y - 12)(y + 3)

y = 12 or  - 3

x = 12² / 18 = 144 / 18 = 8

Finally, lets find the 15th term of the geometric progression

a₁₅ = ar¹⁵⁻¹

a₁₅ = 4 × (12 / 8)¹⁴

a₁₅ = 4 × (3 / 2)¹⁴

a₁₅ = 4 × 291.929260254

a₁₅ = 1167.71704102

a₁₅ = 1167.72

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