To solve this problem, we can use the formula for arithmetic progression where the nth term of the AP is given by:
An = A1 + (n-1)d
where An is the nth term, A1 is the first term, n is the term number, and d is the common difference.
(a) Find the common difference and the first term:
Given: A5 = 18, A10 = -2
To find the common difference, we will subtract A5 from A10:
-2 - 18 = -20
Therefore, the common difference (d) is -20.
To find the first term, we can substitute the values of A5 and d into the formula:
A5 = A1 + (5-1)d
18 = A1 + 4(-20)
18 = A1 - 80
A1 = 18 + 80
A1 = 98
Therefore, the first term (A1) is 98.
(b) Determine the least number of terms which must be added together so that the sum of the progression is negative. Hence, find the sum:
We need to find the number of terms (n) such that the sum of the arithmetic progression is negative. The sum of an arithmetic progression is given by:
Sn = (n/2)(2A1 + (n-1)d)
We want to find the smallest value of n such that Sn is negative. Let's substitute the values of A1, d, and Sn into the formula and solve for n:
Sn = (n/2)(2(98) + (n-1)(-20))
Since we don't know the value of n yet, we'll assume that the sum is negative and solve for n.
Let's say Sn is negative, so Sn < 0:
(n/2)(2(98) + (n-1)(-20)) < 0
Multiplying through by 2 will give:
n(196 - 20n + 20) < 0
Simplifying:
n(216 - 20n) < 0
We have a quadratic inequality, and we can solve this by finding the critical points:
n(216 - 20n) = 0
Setting each factor equal to zero and solving:
n = 0 or 216 - 20n = 0
The first solution, n = 0, is not valid because it is not a positive integer.
Solving the second equation:
216 - 20n = 0
20n = 216
n = 216/20
n = 10.8
The number of terms, n, cannot be a decimal, so we take the next higher integer, which is 11.
Therefore, the least number of terms that must be added together so that the sum of the progression is negative is 11.
To find the sum, we can substitute the values of A1, d, and n into the formula for the sum:
Sn = (n/2)(2A1 + (n-1)d)
S11 = (11/2)(2(98) + (11-1)(-20))
S11 = (11/2)(196 - 200)
S11 = (11/2)(-4)
S11 = -22
Therefore, the sum of the arithmetic progression is -22.