the first and second term of an exponential sequence are respectively the first and third term of a linear sequence the fourth term of the linear sequence is 10 and the sum of it's first five terms is 60 (a)find the first five terms of a linear sequence and the sum of the first n terms
The terms in the bot's answers don't even satisfy the given conditions !!!
first information: term1 of the AS = term1 of the GS, so a = a
term2 of the GS = ar, term2 of the AS = a+d, so a+d = ar, (let that sit)
2nd information:
"the fourth term of the linear sequence is 10 " ...... a + 3d = 10 **
"the sum of it's first five terms is 60" ... (5/2)(2a + 4d) = 60
2a + 4d = 24
a + 2d = 12 ***
subtract *** from ** to get d = -2, which gives us a = 16
back in a+d = ar
16-2 = 16r ...... r = 7/8
our AS is : 16, 14, 12, 10, 8, ...
sum(n) = (n/2(32 + (n-1)(-2))
= (n/2)(34 - 2n) = n(17 - n)
our GS is :16, 14, 49/4, 343/32, ....
All conditions are met,
my answer is correct, the bot is wrong!
Let's break down the problem step by step:
Step 1: Identify the exponential sequence and find the first two terms.
Let's assume the exponential sequence has the general form: a, ar, ar^2, ar^3, ...
The first term of the exponential sequence is "a" and the second term is "ar". These two terms also form the first and third terms of the linear sequence.
Step 2: Find the linear sequence.
Since the first term of the linear sequence is the same as the first term of the exponential sequence, we have:
First term of linear sequence = a
Since the second term of the linear sequence is the same as the third term of the exponential sequence, we have:
Third term of linear sequence = ar
So, the linear sequence can be written as: a, ?, ar, ?, ...
Step 3: Find the fourth term of the linear sequence.
According to the problem, the fourth term of the linear sequence is 10.
Fourth term of linear sequence = 10
Step 4: Find the sum of the first five terms of the linear sequence.
According to the problem, the sum of the first five terms of the linear sequence is 60.
Sum of the first five terms of linear sequence = 60
Step 5: Solve for the first five terms of the linear sequence and the sum of the first n terms.
Using the information we have, let's proceed with the calculations:
First, we need to find the common ratio (r) of the exponential sequence. To do this, we can use the relationship between the first and second terms:
a * r = ar
Dividing both sides by a:
r = ar / a = r
Since we know that r is a common ratio, we can calculate its value as 1.
Now, we can find the other terms of the linear sequence using the identified values:
First term of linear sequence = a
Second term of linear sequence = (ar^2) / r = ar
Third term of linear sequence = ar
Fourth term of linear sequence = 10
Fifth term of linear sequence = (ar^4) / r^2 = ar^2
Now, we can calculate the sum of the first n terms of the linear sequence:
Sum of the first n terms = a + ar + ar^2 + ar^3 + ...
So, the first five terms of the linear sequence are: a, ar, ar, 10, ar^2
And the sum of the first n terms of the linear sequence is: a + ar + ar^2 + ar^3 + ...
To solve this problem, let's break it down step by step:
Step 1: Understand the given information
- The first and second terms of the exponential sequence are respectively the first and third terms of a linear sequence.
- The fourth term of the linear sequence is 10.
- The sum of the first five terms of the linear sequence is 60.
Step 2: Find the common ratio of the exponential sequence
- Since the first term of the exponential sequence corresponds to the first term of the linear sequence, we can say that the first term of the exponential sequence is also the first term of the linear sequence.
- Similarly, since the second term of the exponential sequence corresponds to the third term of the linear sequence, we can say that the second term of the exponential sequence is also the third term of the linear sequence.
- These two terms allow us to find the common ratio (r) of the exponential sequence.
Let's denote the first term of the exponential sequence as a and the common ratio as r. Using the given information:
- The first term of the exponential sequence: a
- The second term of the exponential sequence: a * r
- The third term of the exponential sequence: a * r * r = a * r^2
We know that the second term of the exponential sequence is the third term of the linear sequence:
a * r = a * r^2
By canceling out the 'a' term from both sides of the equation, we get:
r = r^2
This equation simplifies to:
r^2 - r = 0
Factoring out 'r' from the left side:
r(r - 1) = 0
So, the possible values of r are 0 and 1.
Step 3: Determine the value of the common ratio (r)
- If r = 0, then every term of the exponential sequence would be 0, which is not valid since the first term of the exponential sequence is given.
- Therefore, the only possible value for r is 1.
Step 4: Find the linear sequence
- Since r = 1, the exponential sequence is just a sequence of the same number, which means it is a constant sequence.
- This means that the linear sequence is also a constant sequence since the corresponding terms are the same.
- The fourth term of the linear sequence is given as 10, so we can conclude that all the terms in the linear sequence are 10.
Step 5: Find the sum of the first five terms of the linear sequence
- Since all the terms in the linear sequence are 10, the sum of the first five terms is simply 10 + 10 + 10 + 10 + 10 = 50.
So, we have found that the linear sequence is {10, 10, 10, 10, 10} and the sum of its first five terms is 50.
Step 6: Find the sum of the first n terms of the linear sequence
- Since all the terms in the linear sequence are the same, the sum of the first n terms is n times the value of each term.
- Therefore, for the linear sequence we found ({10, 10, 10, 10, 10}), the sum of the first n terms would be 10n.
In summary:
- The first five terms of the linear sequence are {10, 10, 10, 10, 10}.
- The sum of the first n terms of the linear sequence is 10n.
(b)find the first five terms of the exponential sequence and the sum of the first n terms
(a) The first five terms of the linear sequence are 2, 5, 10, 17, and 26. The sum of the first n terms is given by Sn = n(n+1)/2.
(b) The first five terms of the exponential sequence are 2, 8, 32, 128, and 512. The sum of the first n terms is given by Sn = 2^n - 1.