Find two geometric progression having: 2 as second term and 1458 as eight term,
common ratio = (1458/2)^(8-2)= ±3
So the g.p.s are
2/3,2,6,18,54,162,486,1458
or
-2/3,2,-6, ....1458
To find two geometric progressions with 2 as the second term and 1458 as the eighth term, we can use the formula for the nth term of a geometric progression.
The formula for the nth term of a geometric progression is given by:
an = a1 * r^(n-1)
where:
an = nth term
a1 = first term
r = common ratio
Let's denote the two geometric progressions as GP1 and GP2.
For GP1:
Given: a2 = 2 and a8 = 1458
Using the formula, we have:
a2 = a1 * r^(2-1) = a1 * r
a8 = a1 * r^(8-1) = a1 * r^7
Dividing the equations, we get:
a8 / a2 = (a1 * r^7) / (a1 * r) = r^6
Since a8 = 1458 and a2 = 2, we can substitute the values:
1458 / 2 = r^6
Reducing the equation:
729 = r^6
Taking the 6th root of both sides:
r = ∛729 = 9
Substituting r back into the equation a2 = a1 * r, we have:
2 = a1 * 9
Dividing both sides by 9, we get:
a1 = 2/9 = 2/3
Therefore, the first progression GP1 is:
a1 = 2/3 and r = 9
For GP2:
Given: a2 = 2 and a8 = 1458
Using the formula, we have:
a2 = a1 * r^(2-1) = a1 * r
a8 = a1 * r^(8-1) = a1 * r^7
Dividing the equations, we get:
a8 / a2 = (a1 * r^7) / (a1 * r) = r^6
Since a8 = 1458 and a2 = 2, we can substitute the values:
1458 / 2 = r^6
Reducing the equation:
729 = r^6
Taking the 6th root of both sides:
r = ∛729 = 9
Substituting r back into the equation a2 = a1 * r, we have:
2 = a1 * 9
Dividing both sides by 9, we get:
a1 = 2/9 = 2/3
Therefore, the second progression GP2 is:
a1 = 2/3 and r = 9
To find two geometric progressions, we need to determine the common ratio (r) of each progression. The formula for the nth term (An) of a geometric progression is given by:
An = A1 * r^(n-1)
Here, A1 represents the first term of the progression.
Let's assume that the first progression has a common ratio of 'r' and the second progression has a common ratio of 's'.
For the first progression:
Given: A2 = 2
So, A1 * r^(2-1) = 2
A1 * r = 2 -----------(1)
For the second progression:
Given: A8 = 1458
So, A1 * s^(8-1) = 1458
A1 * s^7 = 1458 ----------(2)
Now, we have two equations (equations 1 and 2) with two unknowns (A1 and r). To find the values of A1 and r, we need to solve these equations simultaneously.
Divide equation (2) by equation (1):
(A1 * s^7)/(A1 * r) = 1458/2
Cancel out A1:
s^7/r = 729
Taking the 7th root on both sides:
(s^7/r)^(1/7) = 729^(1/7)
s/r = 3
Now, we know that the common ratio (s/r) equals 3.
Substituting this value in equation (1):
A1 * 3 = 2
Solving for A1:
A1 = 2/3
Therefore, we have the values of A1 and r for the first progression:
A1 = 2/3
r = 3
Now, we can find the terms of the first progression:
A1 = 2/3
A2 = A1 * r = (2/3) * 3 = 2
A3 = A2 * r = 2 * 3 = 6
A4 = A3 * r = 6 * 3 = 18
A5 = A4 * r = 18 * 3 = 54
A6 = A5 * r = 54 * 3 = 162
A7 = A6 * r = 162 * 3 = 486
A8 = A7 * r = 486 * 3 = 1458
So, the first geometric progression is: 2/3, 2, 6, 18, 54, 162, 486, 1458.
Now, let's find the terms of the second progression using a similar approach. We know that the second progression has a common ratio of 'r' as well:
A1 = 2/3
A2 = A1 * r = (2/3) * 3 = 2
A3 = A2 * r = 2 * 3 = 6
A4 = A3 * r = 6 * 3 = 18
A5 = A4 * r = 18 * 3 = 54
A6 = A5 * r = 54 * 3 = 162
A7 = A6 * r = 162 * 3 = 486
A8 = A7 * r = 486 * 3 = 1458
So, the second geometric progression is also: 2/3, 2, 6, 18, 54, 162, 486, 1458.