write the first five terms of a geometric sequence whose first term is 100 and whose third term is less than 30
100r^2 < 30
r < √0.3 = 0.547722
so, one sequence would have r=.5:
100 50 25 12.5 6.25 ...
If r=.5477, we have
100 54.77 29.997 16.429 8.999 ...
The 3rd term is just less than 30.
To find the first five terms of a geometric sequence, we need to calculate the common ratio (r) first. The common ratio is found by dividing any term in the sequence by its previous term.
Given that the first term (a₁) is 100 and the third term (a₃) is less than 30, we can set up the following equation:
a₃ = a₁ * r²
30 = 100 * r²
To solve for r, divide both sides of the equation by 100:
r² = 30/100
r² = 0.3
Now, take the square root of both sides to find r:
r = √0.3
r ≈ 0.55
Now that we have the common ratio (r ≈ 0.55), we can find the first five terms of the geometric sequence.
a₁ = 100 (given)
a₂ = a₁ * r
= 100 * 0.55
= 55
a₃ = a₂ * r
= 55 * 0.55
= 30.25 (less than 30)
a₄ = a₃ * r
= 30.25 * 0.55
≈ 16.63
a₅ = a₄ * r
≈ 16.63 * 0.55
≈ 9.14
Therefore, the first five terms of the geometric sequence are: 100, 55, 30.25, 16.63, 9.14