in a geometric sequence the first term is 2. The 3rd term is the sum of the 1st and 2nd term. Find the ratio, and the 2 terms
a = 2
term(3) = ar^2 = 2r^2
term(2) = ar = 2r
2 + 2r = 2r^2
r^2 - r - 1 = 0
r = (1 ± √5)/2
if r = (1 + √5)/2 , which btw, is the GOLDEN RATIO
the first 2 terms are
2 and 1 + √5
if r = (1 - √5)/2
the first 2 terms are
2 and 1 - √5
check: using my calculator and decimals
terms are 2 , 3.236.. , 5.236..
the third time is the sum of the first two
or
terms are 2, -1.236.. , .7639...
.7639 = 2 - 1.236 , ok
To find the ratio and the two terms in the geometric sequence, we can use the given information.
Let's denote the first term as a₁, the second term as a₂, and the common ratio as r.
We are given that the first term is 2, so we have a₁ = 2.
The third term is the sum of the first and second terms, which means a₃ = a₁ + a₂.
Since a₁ = 2, we substitute it into the equation to get:
a₃ = 2 + a₂.
Now, we know that the terms in a geometric sequence are related by the formula aₙ = a₁ * r^(n-1), where aₙ represents the nth term.
Using this formula, we can express a₃ in terms of a₁ and r:
a₃ = a₁ * r^(3-1).
a₃ = a₁ * r^2.
Substituting a₁ = 2, we have:
2 + a₂ = 2 * r^2.
Simplifying the equation, we get:
a₂ = 2 * r^2 - 2.
To find the ratio, we can divide the second term by the first term:
ratio = a₂ / a₁.
ratio = (2 * r^2 - 2) / 2.
ratio = r^2 - 1.
Therefore, the ratio is r^2 - 1, and the two terms are 2 and (2 * r^2 - 2).