1. Find the sum of the first five terms of the sequence 2/3, 2, 6, 18, .....
2. Is this a geometric sequence? 2x, (4x-2), (6x-4), .....
3. Show the equivalent values of the decimal 0.7575757575..... in fraction form.
4. What is the sum of the geometric sequence 4/3, 2/3, 1/3, 1/6, .....
1. looks like a GS, were a = 2/3, and r = 3
sum(5) = a(r^5 - 1)/(r-1)
= (2/3)(3^5 -1)/2
= .....
2. nope, (4x-2)/(2x) ≠ (6x-4)/(4x-2) for all values of x
it would be true or only x = 1
3. 0.7575757575... = .75 + .0075 + .000075 + ...
a GS where a = .75, r = .01
sum(allterms) = a(1-r)
= .75/.99
= 75/99 = 25/33
check by dividing 25 by 33 on your calculator
4. you try this one, I gave you the formula
1. To find the sum of the first five terms of a sequence, we simply add up the terms. The given sequence is 2/3, 2, 6, 18, .....
The sum would be: 2/3 + 2 + 6 + 18 + ...
2/3 + 2 + 6 + 18 + ... can be written as (2/3) + 2 + 6 + 18(2/3), which simplifies to the common ratio being 2/3.
Now, let's find the sum of the first five terms. We can use the formula for the sum of a geometric sequence:
Sum = (first term * (1 - common ratio^n)) / (1 - common ratio)
In this case, the first term is 2/3, the common ratio is 2/3, and we want to find the sum of the first five terms so n = 5.
Plugging these values into the formula:
Sum = (2/3 * (1 - (2/3)^5)) / (1 - 2/3)
Simplifying this expression will give us the sum of the first five terms.
2. To determine if a sequence is geometric, we need to check if the ratio between consecutive terms is constant. The given sequence is 2x, (4x-2), (6x-4), .....
Let's calculate the ratio between the second and first term:
Ratio = (4x-2) / (2x)
Simplifying:
Ratio = 2(x-1) / x
We can see that the ratio is 2(x-1) / x, which is dependent on x. Therefore, this sequence is not geometric.
3. To express the recurring decimal 0.7575757575... in fraction form, we can use the method of converting repeating decimals to fractions. Let's assign a variable to the decimal, say x:
x = 0.7575757575...
Now, let's multiply both sides of the equation by 100 to remove the repeating decimal:
100x = 75.75757575...
Next, subtract the original equation from the multiplied equation:
100x - x = 75.75757575... - 0.75757575...
Simplifying:
99x = 75
Now, divide both sides by 99 to solve for x:
x = 75/99
To simplify this fraction further, we can divide both the numerator and the denominator by their greatest common divisor, which in this case is 3:
x = (75 ÷ 3) / (99 ÷ 3)
Simplifying:
x = 25/33
So, the recurring decimal 0.7575757575... is equal to 25/33 in fraction form.
4. To find the sum of a geometric sequence with infinite terms, we can use the formula:
Sum = first term / (1 - common ratio)
The given geometric sequence is 4/3, 2/3, 1/3, 1/6, .....
Let's calculate the sum using the formula:
Sum = (4/3) / (1 - 2/3)
Simplifying:
Sum = (4/3) / (1/3)
Dividing by a fraction is the same as multiplying by its reciprocal, so the expression simplifies to:
Sum = (4/3) * (3/1)
Sum = 4
Therefore, the sum of the infinite terms of the geometric sequence 4/3, 2/3, 1/3, 1/6, .... is 4.