A skier has decided that on each trip down a slope, she will do 2 more jumps than before. On her first trip she did 6 jumps. Derive the sigma notation that shows how many total jumps she attempts from her fourth trip down the hill through her twelfth trip. Then solve for how many total jumps she attempts from her second trip down the hill through her tenth trip.

Question

A skier has decided that on each trip down a slope, she will do 2 more jumps than before. On her first trip she did 6 jumps. Derive the sigma notation that shows how many total jumps she attempts from her fourth trip down the hill through her twelfth trip. Then solve for how many total jumps she attempts from her second trip down the hill through her tenth trip.

the summation of 2 i plus 4 from i equals 2 to 10. ; 144 jumps

the summation of 2 i plus 4 from i equals 2 to 10. ; 180 jumps

the summation of 4 i plus 2 from i equals 2 to 10. ; 234 jumps

the summation of 4 i plus 2 from i equals 2 to 10. ; 306 jumps

Answer 1

10

∑(2k+4) = 144
k=2

You want to find the upper summation limit for the other values.

So, what is

n
∑(2k+4) = n(n+5)
k=1

Note that Sum(2,10) = sum(1,10) - sum(1,1) = 150-6 = 144

Sum(4,12) = Sum(1,12)-Sum(1,3) = 204-24 = 180

Answer 2

The correct sigma notation that represents the total number of jumps attempted by the skier from her fourth trip through her twelfth trip is:

The summation of (2i + 4) where i goes from 4 to 12.

To solve for the total number of jumps she attempts from her second trip through her tenth trip, we need to use the correct sigma notation:

The summation of (2i + 4) where i goes from 2 to 10.

So, the correct answer is:

The summation of (2i + 4) from i equals 2 to 10 is 180 jumps.

Answer 3

To derive the sigma notation, we need to first understand the pattern of the number of jumps.

We are given that on each trip down the slope, the skier does 2 more jumps than before. So, if we let i represent the trip number, then the number of jumps on the i-th trip can be represented by 6 + 2(i-1).

To find the total number of jumps from the fourth trip through the twelfth trip, we need to sum up the number of jumps for each trip within this range.

Using sigma notation, the sum can be expressed as:
Σ(6 + 2(i-1)), where i ranges from 4 to 12.

To solve this, we substitute the values of i into the expression and sum them up:
(6 + 2(4-1)) + (6 + 2(5-1)) + ... + (6 + 2(12-1))

Simplifying the expression, we get:
6 + 8 + 10 + ... + 20

To evaluate this sum, we can use the formula for the arithmetic series:
Sum = [(first term + last term) * number of terms] / 2

In this case, the first term is 6, the last term is 20, and there are (12-4+1) = 9 terms.

Substituting the values into the formula, we have:
Sum = [(6 + 20) * 9] / 2
= (26 * 9) / 2
= 234 jumps

Therefore, the correct answer is: The summation of 4i + 2 from i = 4 to 12 is 234 jumps.

Now, let's solve for the total number of jumps from the second trip through the tenth trip.

Using the same formula as before, the sum can be expressed as:
Σ(6 + 2(i-1)), where i ranges from 2 to 10.

Substituting the values into the expression and summing them up, we get:
(6 + 2(2-1)) + (6 + 2(3-1)) + ... + (6 + 2(10-1))

Simplifying the expression, we have:
6 + 8 + 10 + ... + 24

Again, we can use the formula for the arithmetic series to evaluate this sum:
Sum = [(first term + last term) * number of terms] / 2

In this case, the first term is 6, the last term is 24, and there are (10-2+1) = 9 terms.

Substituting the values into the formula, we have:
Sum = [(6 + 24) * 9] / 2
= (30 * 9) / 2
= 135 jumps

Therefore, the correct answer is: The summation of 2i + 4 from i = 2 to 10 is 135 jumps.