The first and second terms of an arithmetic sequence are 50 and 100, respectively.
The first and second terms of a geometric sequence are also 50 and 100,
respectively. The sum of the third terms of these two sequences is:
for the AP, a = 50 d = 50
term3 = 150
for the GP, a = 50, r = 2
term3 = 200
sum of those two terms = 150+200 = 350
Well, isn't that a coincidence? The first and second terms of both the arithmetic and geometric sequences are the same! It's like they're practically twins, just with different tricks up their sleeves.
Now, let's see how they sum up. In the arithmetic sequence, we can find the common difference by subtracting the first term (50) from the second term (100), which gives us a difference of 50.
Since we know the common difference, we can find the third term by adding the common difference twice to the second term. So, 100 + 50 + 50 = 200.
In the geometric sequence, we can find the common ratio by dividing the second term (100) by the first term (50), which gives us a ratio of 2.
Now, let's find the third term of the geometric sequence by multiplying the second term (100) by the common ratio (2). So, 100 x 2 = 200.
Oh, would you look at that! The sum of the third terms of both sequences is 200. It seems like they both have their ways of reaching the same destination, just through different paths.
To find the sum of the third terms of the arithmetic and geometric sequences, we first need to find the common difference (d) and common ratio (r) of each sequence.
For the arithmetic sequence:
Given:
First term (a₁) = 50
Second term (a₂) = 100
The formula for finding the nth term (aₙ) of an arithmetic sequence is:
aₙ = a₁ + (n-1) * d
We can find the common difference (d) by subtracting the first term from the second term and simplifying:
d = a₂ - a₁
d = 100 - 50
d = 50
So, the common difference (d) of the arithmetic sequence is 50.
For the geometric sequence:
Given:
First term (a₁) = 50
Second term (a₂) = 100
The formula for finding the nth term (aₙ) of a geometric sequence is:
aₙ = a₁ * r^(n-1)
We can find the common ratio (r) by dividing the second term by the first term and simplifying:
r = a₂ / a₁
r = 100 / 50
r = 2
So, the common ratio (r) of the geometric sequence is 2.
Now, let's find the third term (a₃) of each sequence:
For the arithmetic sequence:
a₃ = a₁ + (3-1) * d
a₃ = 50 + 2 * 50
a₃ = 50 + 100
a₃ = 150
For the geometric sequence:
a₃ = a₁ * r^(3-1)
a₃ = 50 * 2^(3-1)
a₃ = 50 * 2^2
a₃ = 50 * 4
a₃ = 200
Finally, let's find the sum of the third terms:
Sum = a₃ (arithmetic) + a₃ (geometric)
Sum = 150 + 200
Sum = 350
Therefore, the sum of the third terms of these two sequences is 350.
To find the sum of the third terms of these two sequences, we need to determine the third term for each sequence and then add them together.
For the arithmetic sequence:
We know that the first term is 50 and the second term is 100. Since it is an arithmetic sequence, we can find the common difference by subtracting the first term from the second term: 100 - 50 = 50.
The formula to find the nth term of an arithmetic sequence is: tn = a + (n - 1)d,
where tn is the nth term, a is the first term, n is the position of the term, and d is the common difference.
Using the formula, we can find the third term: t3 = 50 + (3 - 1) * 50 = 50 + 100 = 150.
For the geometric sequence:
We also know that the first term is 50 and the second term is 100. Since it is a geometric sequence, we can find the common ratio by dividing the second term by the first term: 100 / 50 = 2.
The formula to find the nth term of a geometric sequence is: tn = a * r^(n - 1),
where tn is the nth term, a is the first term, n is the position of the term, and r is the common ratio.
Using the formula, we can find the third term: t3 = 50 * 2^(3 - 1) = 50 * 4 = 200.
Now, we can find the sum of the third terms of the arithmetic and geometric sequences by adding them together: 150 + 200 = 350.
Therefore, the sum of the third terms of these two sequences is 350.