Consider the arithmetic sequence 6,10,14,...
Calculate the least number of terms that must be added,if the answer must be more than 2000
To find the least number of terms that must be added, we need to calculate the sum of the arithmetic sequence until the sum is greater than 2000.
First, let's find the common difference (d) of the arithmetic sequence. The common difference can be found by subtracting the first term (a₁) from the second term (a₂):
d = a₂ - a₁ = 10 - 6 = 4
Next, we need to find the sum of the arithmetic sequence. The formula for the sum of an arithmetic sequence is:
Sn = (n/2)(2a₁ + (n-1)d)
where Sn is the sum of the first n terms, a₁ is the first term, and d is the common difference.
We want to find the least number of terms (n) that results in a sum greater than 2000. So we have the inequality:
Sn > 2000
Substituting the formula for Sn, we get:
(n/2)(2a₁ + (n-1)d) > 2000
We can now plug in the values for a₁ and d:
(n/2)(2*6 + (n-1)*4) > 2000
Simplifying further:
(3n + 4(n-1)) > 2000
(3n + 4n - 4) > 2000
7n - 4 > 2000
7n > 2004
n > 2004/7
n > 286.29
Since the number of terms (n) cannot be in decimal form, we need to round up to the nearest whole number. Therefore, the least number of terms that must be added is 287.