The fifth and tenth term of arithmetic progression are 18 and -2 find common difference and determine last number of term that must be added together so that progression is negative
We are given that the fifth term of the arithmetic progression is 18 and the tenth term is -2.
Let's denote the common difference as d.
The fifth term can be expressed as:
a + 4d = 18
The tenth term can be expressed as:
a + 9d = -2
We can subtract the first equation from the second equation to eliminate a:
(a + 9d) - (a + 4d) = -2 - 18
5d = -20
d = -4
Now we can substitute the value of d into either equation to find a:
a + 4(-4) = 18
a - 16 = 18
a = 34
The arithmetic progression is -4, -8, -12, -16, -20, ...
To determine the last term that must be added together so that the progression is negative, we need to find the positive term immediately before it. We can do this by finding the positive term that is closest to zero.
Since the common difference is negative, the progression is decreasing. The progression becomes negative starting from the first term that is less than zero, which is -20. The term before -20 is -16.
Therefore, the last term that must be added together so that the progression is negative is -16.
To find the common difference (d) of an arithmetic progression, we can first use the formula for the nth term (an) of an arithmetic progression:
an = a1 + (n-1)d
where a1 is the first term, n is the term number, and d is the common difference.
Given that the fifth term is 18, we can substitute these values into the formula:
18 = a1 + (5-1)d
18 = a1 + 4d ----- (1)
Similarly, given that the tenth term is -2, we can substitute these values into the formula:
-2 = a1 + (10-1)d
-2 = a1 + 9d ----- (2)
We now have a system of two equations with two variables (a1 and d). To solve this system, we can subtract equation (2) from equation (1):
18 - (-2) = (a1 + 4d) - (a1 + 9d)
20 = -5d
Now we can solve for d:
-5d = 20
d = -20/5
d = -4
So, the common difference of the arithmetic progression is -4.
To determine the last number of the term that must be added together so that the progression is negative, we need to find the number of terms (n) before the sequence becomes negative. We can use the formula for the nth term:
an = a1 + (n-1)d
When the last term in the progression becomes negative, an will be negative. Therefore, we can set an < 0 and solve for n:
a1 + (n-1)d < 0
Substituting the values, we get:
18 + (n-1)(-4) < 0
Simplifying the equation:
18 - 4n + 4 < 0
22 - 4n < 0
Solving for n:
-4n < -22
n > -22 / (-4)
n > 11/2
Since n represents the number of terms in the sequence, it must be a whole number. Therefore, the last number of terms that must be added together so that the progression is negative is 12.