24. Four friends attempted to write the explicit expression for the nth term of the sequence 2, 5, 10, 17, ….
24. Four friends attempted to write the explicit expression for the nth term of the sequence 2, 5, 10, 17, ….
If n represents the set of counting numbers, who wrote the correct expression
If n represents the set of counting numbers, who wrote the correct expression?
A. Austin
B. Kaylee
C. Noah
D. Zoey
2+3 = 5
5+5 = 10
10+7 = 17
3, 5, 7 are the first three prime numbers after [1 and] 2. Let p1 = 3, p2=5, p3=7, etc. It is logical to assume that the next terms in the sequence are
17+11 = 28
28+13 = 41
etc.
so that we have
2+p1 = t1 (first term in sequence)
t1+p2 = t2
t2+p3 = t3
...
t(n-1)+pn = tn
or...
2+p1+p2+...+pn = tn
This last is the general term sought.
Or
did you notice that each of the given numbers is one more than the perfect square, that is
2 = 1^2 + 1
5 = 2^2 +1
10 = 3^2 + 1
17 = 4^2 + 1
thus:
term(n) = n^2 + 1
To determine who wrote the correct expression, let's examine the given sequence:
2, 5, 10, 17, ...
To find the explicit expression for the nth term, we can observe the pattern in the sequence.
The difference between consecutive terms is increasing by 1 each time:
5 - 2 = 3
10 - 5 = 5
17 - 10 = 7
This suggests that the sequence is formed by adding consecutive odd numbers starting from 1.
To confirm this, we can check each friend's expression:
A. Austin: 2n + 1
B. Kaylee: n^2 + 1
C. Noah: 2n^2 + 1
D. Zoey: 2n^2 - n + 1
We can now substitute n = 1 into each expression and see which one produces the first term of the sequence, which is 2.
A. Austin: 2(1) + 1 = 3
B. Kaylee: (1)^2 + 1 = 2
C. Noah: 2(1)^2 + 1 = 3
D. Zoey: 2(1)^2 - 1 + 1 = 2
From the results, we can see that Kaylee's expression, n^2 + 1, correctly produces the first term of the sequence. Therefore, the answer is B. Kaylee.
To determine who wrote the correct expression for the nth term of the sequence 2, 5, 10, 17, ..., we can analyze the pattern in the sequence.
The given sequence does not appear to follow a simple arithmetic pattern where each term is obtained by adding a fixed number to the previous term. Instead, it seems that the differences between consecutive terms are increasing. Let's calculate the differences:
Difference between the 1st and 2nd terms: 5 - 2 = 3
Difference between the 2nd and 3rd terms: 10 - 5 = 5
Difference between the 3rd and 4th terms: 17 - 10 = 7
We can observe that the differences form their own sequence: 3, 5, 7, ...
We can notice that these differences are consecutive odd numbers starting from 3. This suggests that the nth term of the sequence can be obtained by adding the sum of the first n odd numbers to 2.
Now, let's find the sum of the first n odd numbers. The sum of the first n odd numbers can be calculated using the formula: n^2.
Therefore, the nth term of the sequence can be expressed as: 2 + n^2.
Now, let's analyze the expressions given by the four friends:
A. Austin: Austin did not provide an expression for the nth term of the sequence.
B. Kaylee: Kaylee did not provide an expression for the nth term of the sequence.
C. Noah: Noah did not provide an expression for the nth term of the sequence.
D. Zoey: Zoey provided the expression n^2 + 2.
From our analysis, we determined that the correct expression for the nth term of the sequence is 2 + n^2, which matches Zoey's expression.
Therefore, the correct answer is D. Zoey.