ar^2 = a+12
ar^3 = ar+36
Find four numbers that form a geometric progression such that the third term is greater than the first by 12 and the fourth is greater than the second by 36.
3 answers
The n-th term of a geometric sequence with initial value a and common ratio r is given by:
an = a * r ^ ( n - 1 )
a is the first term
r is common ratio
In this case:
a1 = a * r ^ ( 1 - 1 ) = a * r ^ 0 = a * 1 = a
a2 = a * r ^ ( 2 - 1 ) = a * r ^ 1 = a * r
a3 = a * r ^ ( 3 - 1 ) = a * r ^ 2
a4 = a * r ^ ( 4 - 1 ) = a * r ^ 3
The third term is greater than the first by 12 mean:
a3 = a1 + 12
The fourth is greater than the second by 36 mean:
a4 = a2 + 36
So:
a3 = a1 + 12
a * r ^ 2 = a + 12
a4 = a2 + 36
a * r ^ 3 = a * r + 36
Now you must solve system of 2 equations with 2 unknow:
a * r ^ 2 = a + 12
a * r ^ 3 = a * r + 36
a * r ^ 2 = a + 12 Subtract a to both sides
a * r ^ 2 - a = a + 12 - a
a * r ^ 2 - a = 12
a * ( r ^ 2 - 1 ) = 12
a * r ^ 3 = a * r + 36 Subtract a * r to both sides
a * r ^ 3 - a * r = a * r + 36 - a * r
a * r ^ 3 - a * r = 36
a * r * ( r ^ 2 - 1 ) = 36
Your system become:
a * ( r ^ 2 - 1 ) = 12
a * r * ( r ^ 2 - 1 ) = 36
a * ( r ^ 2 - 1 ) = 12 Divide both sides by a
r ^ 2 - 1 = 12 / a
a * r * ( r ^ 2 - 1 ) = 36 Divide both sides by a * r
r ^ 2 - 1 = 36 / a * r
r ^ 2 - 1 = r ^ 2 - 1
12 / a = 36 / a * r Multiply both sides by a
12 * a / a = 36 * a / a * r
12 = 36 / r Multiply both sides by r
12 * r = 36 * r / r
12 r = 36 Divide both sides by 12
r = 36 / 12
r = 3
Replace this value in equation:
a * ( r ^ 2 - 1 ) = 12
a * ( 3 ^ 2 - 1 ) = 12
a * ( 9 - 1 ) = 12
a * 8 = 12
8 a = 12 Divide both sides by 8
a = 12 / 8
a = 4 * 3 / ( 4 * 2 )
a = 3 / 2
a = 3 / 2 , r = 3 so:
a1 = a
a1 = 3 / 2
a2 = a * r
a2 = ( 3 / 2 ) * 3 = 9 / 2
a3 = a * r ^ 2
a3 = ( 3 / 2 ) * 3 ^ 2 = ( 3 / 2 ) * 9 = 27 / 2
a4 = a * r ^ 3
a4 = ( 3 / 2 ) * 3 ^ 3 = ( 3 / 2 ) * 27 = 81 / 2
Your geometric progression:
3 / 2 , 9 / 2 , 27 / 2 , 81 / 2
Proof:
a3 = a1 + 12
27 / 2 = 3 / 2 + 12
27 / 2 = 3 / 2 + 12 * 2 / 2
27 / 2 = 3 / 2 + 24 / 2
a4 = a2 + 36
81 / 2 = 9 / 2 + 36
81 / 2 = 9 / 2 + 36 * 2 / 2
81 / 2 = 9 / 2 + 72 / 2
an = a * r ^ ( n - 1 )
a is the first term
r is common ratio
In this case:
a1 = a * r ^ ( 1 - 1 ) = a * r ^ 0 = a * 1 = a
a2 = a * r ^ ( 2 - 1 ) = a * r ^ 1 = a * r
a3 = a * r ^ ( 3 - 1 ) = a * r ^ 2
a4 = a * r ^ ( 4 - 1 ) = a * r ^ 3
The third term is greater than the first by 12 mean:
a3 = a1 + 12
The fourth is greater than the second by 36 mean:
a4 = a2 + 36
So:
a3 = a1 + 12
a * r ^ 2 = a + 12
a4 = a2 + 36
a * r ^ 3 = a * r + 36
Now you must solve system of 2 equations with 2 unknow:
a * r ^ 2 = a + 12
a * r ^ 3 = a * r + 36
a * r ^ 2 = a + 12 Subtract a to both sides
a * r ^ 2 - a = a + 12 - a
a * r ^ 2 - a = 12
a * ( r ^ 2 - 1 ) = 12
a * r ^ 3 = a * r + 36 Subtract a * r to both sides
a * r ^ 3 - a * r = a * r + 36 - a * r
a * r ^ 3 - a * r = 36
a * r * ( r ^ 2 - 1 ) = 36
Your system become:
a * ( r ^ 2 - 1 ) = 12
a * r * ( r ^ 2 - 1 ) = 36
a * ( r ^ 2 - 1 ) = 12 Divide both sides by a
r ^ 2 - 1 = 12 / a
a * r * ( r ^ 2 - 1 ) = 36 Divide both sides by a * r
r ^ 2 - 1 = 36 / a * r
r ^ 2 - 1 = r ^ 2 - 1
12 / a = 36 / a * r Multiply both sides by a
12 * a / a = 36 * a / a * r
12 = 36 / r Multiply both sides by r
12 * r = 36 * r / r
12 r = 36 Divide both sides by 12
r = 36 / 12
r = 3
Replace this value in equation:
a * ( r ^ 2 - 1 ) = 12
a * ( 3 ^ 2 - 1 ) = 12
a * ( 9 - 1 ) = 12
a * 8 = 12
8 a = 12 Divide both sides by 8
a = 12 / 8
a = 4 * 3 / ( 4 * 2 )
a = 3 / 2
a = 3 / 2 , r = 3 so:
a1 = a
a1 = 3 / 2
a2 = a * r
a2 = ( 3 / 2 ) * 3 = 9 / 2
a3 = a * r ^ 2
a3 = ( 3 / 2 ) * 3 ^ 2 = ( 3 / 2 ) * 9 = 27 / 2
a4 = a * r ^ 3
a4 = ( 3 / 2 ) * 3 ^ 3 = ( 3 / 2 ) * 27 = 81 / 2
Your geometric progression:
3 / 2 , 9 / 2 , 27 / 2 , 81 / 2
Proof:
a3 = a1 + 12
27 / 2 = 3 / 2 + 12
27 / 2 = 3 / 2 + 12 * 2 / 2
27 / 2 = 3 / 2 + 24 / 2
a4 = a2 + 36
81 / 2 = 9 / 2 + 36
81 / 2 = 9 / 2 + 36 * 2 / 2
81 / 2 = 9 / 2 + 72 / 2
Nice work. I'd have changed things a bit here:
Your system become:
a * ( r ^ 2 - 1 ) = 12
a * r * ( r ^ 2 - 1 ) = 36
Divide the bottom by the top to get
r = 36/12 = 3
a*8 = 12
a = 3/2
Your system become:
a * ( r ^ 2 - 1 ) = 12
a * r * ( r ^ 2 - 1 ) = 36
Divide the bottom by the top to get
r = 36/12 = 3
a*8 = 12
a = 3/2