The second and fifth terms of a geometric progression(G.P. ) are 1 and 1/8 respectively find the first term and eight term.
a1 = the first term
r = common ratio
The n-th term of a geometric sequence:
xn = a ∙ r ⁿ ⁻ ¹
x2 = a1 ∙ r = 1
x5 = a1 ∙ r⁴ = 1 / 8
x5 / x2 =
a r⁴ / a r = ( 1 / 8 ) / 1
r³ = 1 / 8
r = ∛( 1 / 8 ) = ∛1 / ∛8 = 1 / 2
a1 ∙ r = 1
a1 = 1 / r = 1 / ( 1 / 2 ) = 2
a8 = a1 ∙ r⁷ = 2 ∙ ( 1 / 2 )⁷ = 2 ∙ 1⁷ / 2⁷ = 2 ∙ 1 / 128 = 1 / 64
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Well, well, well, we have ourselves a geometric progression conundrum, don't we? Let's see if we can unravel this puzzle!
Now, in a geometric progression, each term is found by multiplying the previous term by a constant value called the common ratio. So, let's call this mysterious common ratio "r".
We know that the second term is 1, so that means the first term multiplied by "r" gives us 1. Therefore, our first term is 1 divided by "r". Got it?
Now, moving on to the fifth term, which is 1/8. We can find this by multiplying the first term (1/r) by "r" four times. So, we have (1/r) x (r^4) = 1/8. Now, we just need to solve this equation to find the value of "r".
Once we've found the value of "r", we can then substitute it back into the first term expression (1/r) to find the actual value of the first term. And voila, the mystery is solved!
As for the eighth term, well, now that we have the first term and the common ratio, we can simply find it by multiplying the first term by "r" seven times. So, take the first term (1/r), multiply it by "r", then multiply that result by "r" again, and so on, all the way up to the eighth term.
Please feel free to follow these steps and let me know if you need any further assistance or if I need to break out my clown shoes for some extra laughs!
To find the first term and eighth term of a geometric progression (G.P.), we need to use the given information that the second term is 1 and the fifth term is 1/8.
The general formula for the nth term of a geometric progression is given by:
aₙ = a₁ * r^(n-1)
Where aₙ is the nth term, a₁ is the first term, r is the common ratio, and n is the term number.
Step 1: Finding the common ratio (r)
We can calculate the common ratio (r) using the following formula:
r = (n-2nd term) / (n-1st term)
Substituting the given values:
r = (1/8) / 1
r = 1/8
Step 2: Finding the first term (a₁)
We can now use the formula for the second term (a₂) to find the first term (a₁):
1 = a₁ * (1/8)^(2-1)
1 = a₁ * (1/8)
Multiplying both sides by 8:
8 = a₁
Therefore, the first term (a₁) is 8.
Step 3: Finding the eighth term (a₈)
We can use the formula for the eighth term (a₈) using the common ratio and the first term we obtained:
a₈ = 8 * (1/8)^(8-1)
a₈ = 8 * (1/8)^7
Simplifying the exponent:
a₈ = 8 * (1/8)^6 * (1/8)
a₈ = 8 * (1/64) * (1/8)
a₈ = 1/8
Therefore, the eighth term (a₈) is 1/8.
In conclusion, the first term is 8 and the eighth term is 1/8.