The second and fifth terms of a geometric progression (G.P) are 1 and ⅛ respectively. Find the:
A, common ratio;
B, first term;
C, eighth term.
term(2) = ar = 1
term(5) = ar^4 = 1/8
divide ....
ar^4/(ar) = 1/8 / 1
r^3 = 1/8
r = 1/2
sub into ar = 1 to find a
after that, use your definitions for term(8)
Pls how did you get 1/2
Please I don't understand
A. Well, the common ratio is just like your favorite jeans – it's the thing that connects all the terms! To find it, we simply divide the second term by the first term. So in this case, the common ratio A = 1/8 divided by 1 equals... drumroll please... ⅛!
B. Now let's find the first term, shall we? Since we know the common ratio, we can use it to go back one step. But because math is like a game of reverse limbo – the lower you go, the more complicated it gets – we'll use a little trick. We'll divide the second term by the common ratio. So B = 1 divided by ⅛... which gives us... 8! *ta-dah!*
C. Finally, let's calculate the eighth term. To do this, we can use the formula for the nth term of a geometric progression: T(n) = T(1) × r^(n-1). In this case, T(1) (the first term) is 8, and r (the common ratio) is ⅛. Plugging these values into the formula, we get T(8) = 8 × (⅛)^(8-1). Simplifying that mess, we find the eighth term to be 1/64. That's definitely a small number, but don't worry – size doesn't matter in this case! 🤡
To find the common ratio (A), we can use the formula for the nth term of a geometric progression:
nth term = first term * (common ratio) ^ (n-1)
We have two pieces of information:
- The second term is 1
- The fifth term is 1/8
Using the formula above, we can form two equations:
1 = first term * (common ratio) ^ (2-1)
1/8 = first term * (common ratio) ^ (5-1)
Let's solve these equations to find the values of A, B, and C.
First, let's find the common ratio (A):
1 = B * A^(2-1)
1/8 = B * A^(5-1)
Simplifying the equations:
1 = B * A
1/8 = B * A^4
Since the second equation has A^4, we can rewrite it as:
1/8 = B * (A^2)^2
Taking the square root of both sides:
1/2 = A^2
Now, take the square root of 1/2:
√(1/2) = A
Simplifying the result:
A = √0.5 = 0.7071 (approximately)
Now that we have the common ratio (A), we can find the first term (B). Using the equation:
1 = B * A
Substituting the value of A:
1 = B * 0.7071
Solving for B:
B = 1 / 0.7071 = 1.414 (approximately)
Therefore, the common ratio (A) is approximately 0.7071, and the first term (B) is approximately 1.414.
To find the eighth term (C), we can use the formula for the nth term:
C = B * A^(8-1)
C = 1.414 * (0.7071)^(7)
C = 1.414 * 0.1778
C = 0.2519 (approximately)
Therefore, the eighth term (C) is approximately 0.2519.