Question

Consider the arithmetic sequence -1/5;-11/5;21/5;...
1. Calculate the sum of the first 30 terms of this sequence.
Calculate the least number of terms that must be added, if the answer must be less than -100

Answer 1

To find the sum of an arithmetic sequence, we can use the formula:

Sn = n/2 * (a1 + an)

where Sn is the sum of the first n terms, a1 is the first term, and an is the nth term.

In this case, the first term is -1/5 and the common difference is (-11/5) - (-1/5) = -10/5 = -2.

To find the 30th term, we can use the formula for the nth term of an arithmetic sequence:

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.

In this case, n = 30, a1 = -1/5, and d = -2:

a30 = (-1/5) + (30 - 1) * (-2)
= -1/5 + 29 * (-2)
= -1/5 - 58
= -1/5 - 290/5
= -291/5

Now we can calculate the sum of the first 30 terms:

S30 = 30/2 * (-1/5 + (-291/5))
= 15 * (-1/5 - 291/5)
= 15 * (-292/5)
= -4380/5
= -876

Therefore, the sum of the first 30 terms of this sequence is -876.

To find the least number of terms that must be added to get a sum less than -100, we can set up an inequality:

Sx < -100

We can solve this to find the value of x:

x/2 * (-1/5 + (a1 + (x - 1) * d)/5) < -100
x/2 * (a1 + (a1 + (x - 1) * d))/5 < -100
x * (a1 + (a1 + (x - 1) * d))/10 < -100

In this case, a1 = -1/5 and d = -2:

x * (-1/5 + (-1/5 + (x - 1) * (-2))/5)/10 < -100
x * (-1/5 + (-1/5 - 2(x - 1))/5)/10 < -100
x * (-1/5 - (1/5 + 2(x - 1))/5)/10 < -100
x * (-(1 + 1 + 2x - 2)/5)/10 < -100
x * (-(2x - 2)/5)/10 < -100
-(2x - 2)x/50 < -100

Multiplying both sides by -50 to eliminate the fraction and flipping the direction of the inequality:

(2x - 2)x > 5000

2x^2 - 2x > 5000

Rearranging the equation:

2x^2 - 2x - 5000 > 0

This is a quadratic inequality. We can solve it by finding the roots of the equation and determining the range of x that makes the inequality true.

Solving for x:

2x^2 - 2x - 5000 = 0

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac))/(2a)

In this case, a = 2, b = -2, and c = -5000:

x = (-(-2) ± √((-2)^2 - 4 * 2 * -5000))/(2 * 2)
= (2 ± √(4 + 40000))/4
= (2 ± √40004)/4
= (2 ± 200.01)/4

The approximate values of the roots are:

x ≈ (2 + 200.01)/4 ≈ 50.0025
x ≈ (2 - 200.01)/4 ≈ -49.9975

Since we're looking for the least number of terms that makes the sum less than -100, we need to round up to the nearest integer:

The least number of terms that must be added is 51.