Answer:
Explanation:
Let's find the first six terms of the original sequence for each of the given cases:
a. The first term of the original sequence is 3:
In this case, we can start with the first term as 3 and use the differences provided to find the subsequent terms:
1st term: 3 (given)
2nd term: 3 + 2 = 5
3rd term: 5 + 5 = 10
4th term: 10 + 8 = 18
5th term: 18 + 11 = 29
6th term: 29 + 14 = 43
So, the first six terms of the original sequence are 3, 5, 10, 18, 29, and 43.
b. The sum of the first two terms in the original sequence is 12:
In this case, we know that the sum of the first two terms is 12. Let's call the first term "a" and the second term "b." So:
a + b = 12
We also have the differences:
2 (from a to b)
5 (from b to the next term)
8
11
14
Let's solve for a and b:
From the given differences:
a + 2 = b
b + 5 = (b + 2) + 5 = b + 7
(b + 2) + 8 = b + 10
(b + 10) + 11 = b + 21
(b + 21) + 14 = b + 35
Now, let's check when a + b = 12:
a + (a + 2) = 12
2a + 2 = 12
2a = 12 - 2
2a = 10
a = 5
Now that we have a = 5, we can find b:
a + 2 = b
5 + 2 = b
b = 7
So, the first two terms of the original sequence are 5 and 7. We can use these values to find the subsequent terms, as shown in part (a).
c. The fifth term in the original sequence is 34:
In this case, we need to find the first term "a" and then use the differences to find the fifth term. Let's call the first term "a":
a + 2 = 34 (since the fifth term is 34)
a = 34 - 2
a = 32
Now that we have a = 32, we can find the second term:
a + 2 = b
32 + 2 = b
b = 34
Now, using these values for a and b, we can find the subsequent terms as shown in part (a):
1st term: 32
2nd term: 34
3rd term: 34 + 2 = 36
4th term: 36 + 5 = 41
5th term: 41 + 8 = 49
6th term: 49 + 11 = 60
So, the first six terms of the original sequence are 32, 34, 36, 41, 49, and 60.