To solve the first one, let's try to first find the 26th number. Notice there is a pattern:
To get to the second number from the first, we take one step and go forward 4.
To get to the third number from the first, we take 2 steps and go forward 8.
To get to the fourth pattern from the first, we take 3 steps and go forward 12.
See the pattern? To get to the nth term from the first, we take n - 1 steps and go forward 4(n - 1). (It is multiplied by 4 because the difference between two neighbouring numbers is 4.) We want to get to the 26th term, so we take 25 steps and go forward 4 * 25 = 100. So, the 26th term is 100 + 7 = 107.
To find the sum of 7 + 11 + 13 + 17 + 21 + ...... + 99 + 103 + 107, imagine pairing the 1st with the 26th, the 2nd with the 25th, the 3rd with the 24th, and so on. There will be 26 / 2 = 13 pairs with each one summing to 7 + 107 = 114. So, the sum is 13 * 114 = 1482.
To solve the second one, first note that the weird looking symbol just means summation. The n = 1 on the bottom tells us what number to plug into the expression on the side, 4n + 1. Then, we repeat with 2, 3, 4, 5, all the way up to the number on the top, 32. Then, we sum all of these values. So, what we want is 5 + 9 + 13 + 17 + ..... + 121 + 125 + 129. Pair them up again to get 32 / 2 = 16 pairs with each one having a sum of 5 + 129 = 134. Multiply to get 16 * 134 = 1600 + 480 + 64 = 2080 + 64 = 2114.
To solve the third one, we already know what we want: 1 + 2 + 3 + 4 + 5 + ..... + 199 + 200. Pair these up to get 200 / 2 = 100 pairs, with each one summing to 1 + 200 = 201. So, the sum is 20100.
The answers are:
Hope this helped,
For each one, we check to see if any part of the interval is below the x-axis. If so, we do not check that one. For (-2, 0), we can tell that the whole function is negative, not positive. (An example is (-1, -1.475), where the y-coordinate is negative.) For (0, 4), by inspection, there is no part of the graph below the x-axis. For (4, ∞), we see that something like (5, y) is already way under the x axis, so this is not a choice.
For (-∞, -2), we can see that all of it is above the x-axis. So, this is an option. For (2, 2.5), this includes the interval at the topmost part of the second curve. This is above the x-axis, so this is a choice. Finally, for (-1.5, -1), this is the bottommost part of the first curve, which is below the x-axis. So, this does not work.
So, our final answers are (0, 4), (-∞, -2), and (2, 2.5). (2nd, 4th, and 5th choices.)
it is a right isosceles triangle. Notice the formula is the same as the double angle formula sin(2x) = 2sin(x)cos(x). Here 2x = A and x = B = C. Without proof I show that sin(90) - 2sin(45)cos(45) = 0 since 1 -2(1/sqrt2)(1/sqrt2) = 0.
But a proof shows that it does not have to be a right triangle -- just any isosceles triangle
sinA = 2sinBcosC
sinA = sin(180 -(B + C)) = sin(B + C)
sinA = sinBcosC + cosBsinC
2sinBcosC = sinBcosC + cosBsinC
sinBcosC = cosBsinC
sinB/cosB = sinC/cosC
tanB = tanC
B = C
the triangle is isosceles
Im the creator of this question and wanted to solve it for anyone who needs help with a similar question because I didn't really understand Melody's answer. This was on of my practice questions on my study guide and found out later that my teacher said we didn't need to use a table of values so I solved it with my sister's help. :)
So it's a lot like algerbra except a lot more annoying in my opinion because i struggle with factoring. So you start by setting everything etting everything to zero.
Subtract x and ad 2 to both sides to set the equation equal to 0
\(x = 5, 2\)
Really hope this helps anybody struggling with this. Can't 100% confirm this is right because this is my own work BUT this is how I got my answer for this question.
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Lets break down the logic and axtually take some sense this is place for people with school SHE GAVE A VALID ANSWER we don't need to tell people to there garbage at given answers if it's funny to you it isn't funny to other people they aren't all whiz's but try to contribute we don't need to facilate jerks who are rude to innocent people.
Didn't insult anyone. Just pointed out that explanations of how answers are derived is better than just posting an answer out of the blue. GA is a bit acerbic...but she posts good solutions to often very difficult questions......I got no problem with that....take it with a grain a of salt.... Pretty sure GA has laughed at some of my solutions too......don't be a snowflake. Grow a thicker skin....cowgirl or cowboy up!