Each term in a sequence of numbers, except for the 1st term, is 2 less than the square root of the previous term.If the 3rd term of this sequence is 1, what is the first term?
Let's start by labeling the terms in the sequence. Let's call the first term 'a', the second term 'b', the third term 'c', and so on.
We know that each term, except for the first term, is 2 less than the square root of the previous term. So, we can write the relationship between the terms as follows:
b = sqrt(a) - 2
c = sqrt(b) - 2
Given that the third term (c) is 1, we can substitute it into the equation:
1 = sqrt(b) - 2
To solve for b, we can add 2 to both sides of the equation:
1 + 2 = sqrt(b)
This simplifies to:
3 = sqrt(b)
To isolate b, we square both sides of the equation:
3^2 = b
Therefore, b = 9.
Now, we can substitute the value of b into the equation for the first term (a):
9 = sqrt(a) - 2
To solve for a, we can add 2 to both sides of the equation:
9 + 2 = sqrt(a)
This simplifies to:
11 = sqrt(a)
To isolate a, we square both sides of the equation:
11^2 = a
Therefore, a = 121.
Thus, the first term of the sequence is 121.
To solve this problem, we can work backwards from the given information.
Let's define the terms of the sequence as follows:
- The first term is represented by T1.
- The second term is represented by T2.
- The third term is represented by T3.
We are told that each term in the sequence, starting from the second term, is 2 less than the square root of the previous term. This can be represented mathematically as:
Tn = √(Tn-1) - 2
Given that the third term (T3) is 1, we can substitute the values into the equation:
T3 = 1
T2 = √(T3) - 2 = √1 - 2 = 1 - 2 = -1
T1 = √(T2) - 2 = √(-1) - 2
However, taking the square root of a negative number is not possible within the real number system. This indicates that there is a mistake or inconsistency in the problem statement, as the terms of the sequence would not be valid real numbers.
Therefore, there is no valid value for the first term (T1) that satisfies the given conditions.
121
according to your wording of the problem,
term(n) = √term(n-1) - 2
or
√term(n-1) = term(n) + 2
square both sides
term(n-1) = term(n) ^2 + 4term(n) + 4
for n = 3, term(3) = 1
term(2) = 1^2 + 4(1) + 4
= 9
term(4) = √term(3) - 2 = -1
so we can't go any higher than term(4)
since term(5) = √-1 - 2 , which is undefined in the set of real numbers.
if we follow the rule,
term(1) = term(2) ^2 + 4term(2) + 4
= 81 + 4(9) + 4 = 121
But you said it does not follow the rule for term(1), so I guess term(1) shall remain a mystery since you state no rule for term(1)
Check your typing