i am stuck on this one math problem. The question is to write a rule for the pattern (it is a section on sequences)
The pattern is -5,10,-15,20,-25. What is the rule? A sub n = ?
termn = (5n)(-1)^n
The signs are alternating and each term is
a multiple of 5. The rule would be
(-1)^n(5*n) where n are the integers
1 through 5. To show you that this is true I will evaluate the expression for
each value of n 1 through 5
n=1 (-1)^1*5*1=-1(5)=-5
n=2 (-1)^2*5*2=1(10)=10
n=3 (-1)^3*5*3=-1(15)=-15
n=4 (-1)^4*5*4=1(20)=20
n=5 (-1)^5*5*5=-1(25)=-25
To find the rule for this pattern, we need to identify the relationship between the given numbers. In this case, we can see that the numbers alternate between positive and negative, with increments of 5.
To write the rule, we can use the concept of arithmetic sequences. In an arithmetic sequence, the difference between consecutive terms is constant.
In this given pattern, we can observe that the terms increase by 5 when they are positive and decrease by 5 when they are negative. So, we can say that the common difference in this sequence is -5 and 5 alternately.
Let's split the sequence into two parts: one for the positive terms and another for the negative terms.
Positive terms: 10, 20
Negative terms: -5, -15, -25
We can see that the positive terms can be calculated using the formula: a sub n = a sub 1 + (n - 1)d, where "a sub n" represents the nth term, "a sub 1" is the first term, "n" is the position of the term, and "d" is the common difference.
For the positive terms:
a sub n = 10 + (n - 1)5
= 10 + 5n - 5
= 5n + 5
Similarly, for the negative terms, we have a common difference of -5. So, the formula becomes:
a sub n = -5 + (n - 1)(-5)
= -5 - 5n + 5
= -5n
Combining the two formulas, we can write the rule for the entire pattern:
a sub n = 5n + 5, for the even positions (positive terms)
a sub n = -5n, for the odd positions (negative terms)
In conclusion, the rule for the given pattern is a sub n = 5n + 5 for even values of n and a sub n = -5n for odd values of n.