Use Gauss's approach to find the following sum (do not use formulas):5+11+17+23...+83
Answers
Denote the sum by S. So
S = 5 + 11 + 17 + 23 + ... + 83
There's a constant difference of 6 between consecutive terms in S, so the 3 terms before 83 are 77, 71, and 65. So
S = 5 + 11 + 17 + 23 + ... + 65 + 71 + 77 + 83
Gauss's approach involves inverting the sum:
S = 83 + 77 + 71 + 65 + ... + 23 + 17 + 11 + 5
If we add terms in the same position in the sums, we get
2S = (5 + 83) + (11 + 77) + ... + (77 + 11) + (83 + 5)
and we notice that each grouped term on the right gives a total of 88. So the right side consists of several copies n of 88, which means
2S = 88n
and dividing both sides by 2 gives
S = 44n
Now it's a matter of determining how many copies get added. The terms in the sum form an arithmetic progression that follows the pattern
11 = 5 + 6
17 = 5 + 2*6
23 = 5 + 3*6
and so on, up to
83 = 5 + 13*6
so n = 13, which means the sum is S = 44*13 = 572.
Final answer:
To find the sum of the given arithmetic series, we can use Gauss's approach by finding the number of terms and then calculating the sum using the formula for the sum of an arithmetic series.
Explanation:
To find the sum of the given series, we can use Gauss's approach. The series is an arithmetic progression with a common difference of 6. We can find the number of terms in the series using the formula for the nth term of an arithmetic sequence and then use the formula for the sum of an arithmetic series to find the sum.
- Find the number of terms (n):
- The first term (a) is 5, and the common difference (d) is 6.
- Find the last term (l) using the formula: l = a + (n - 1)d
- Substitute the values and solve for n: 83 = 5 + (n - 1)6
- Simplify and solve for n: n = 15
- Find the sum (S):
- The formula for the sum of an arithmetic series is: S = (n/2)(a + l)
- Substitute the values and solve for S: S = (15/2)(5 + 83)
- Calculate the sum: S = 15(44)
- Simplify the sum: S = 660
Learn more about Gauss's here:
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42Solve the equation: 8 =a) z = 10b) z = 44c) z=3d) z=7
Answers
Answer:
Step-by-step explanation:
5
x + y = 5 => x = 5 - y
y - 4 = - x
=> y - 4 = - ( 5 - y)
=> y - 4 = - 5 + y
=>4 = 5 , not true
So, no solution => system is inconsistent
6
option b
Explanation just as above.
Answers
Answer:
149.86 cm
Step-by-step explanation:
Answers
Answer:
The contestant should not play.
Step-by-step explanation:
As per the given question, their current winning is $1 million.
The probability of the guessing to be true is 50% = .
There is also a possibility of 50% to be wrong, which can reduced the winning amount to $500,000 that is the half of the current amount.
Hence, the contestant should not play.
Answers
Answer:
1. Y
2. N
3. N
4. N
Step-by-step explanation:
Let's use the second equation, since it seems to be easier to use.
To check if an ordered pair is a solution, plug it in to the equation.
1. --> Y
2. --> N
3. --> N
4. --> N
Edit : The 4th equation doesn't work for the first equation, whereas the first one still does.
Answer:
(-2,-6)
Step-by-step explanation:
-9x +2y = 6
5x - 3y = 8
1) Make one of the coefficients the same - y.
-9x +2y = 6 * 3
5x - 3y = 8 * 3
-27x +6y = 18
10x - 6y = 16
2) Add the new equations.
(-27x +6y = 18) + (10x - 6y = 16)
(-27x +6y) + (10x - 6y) = 18 + 16
-17x = 34
3) Divide to find the value of x
-17x = 34
x = 34/-17
x = -2
4) Substitute x into either equation to find the value of y.
-9(-2) +2y = 6
18 +2y = 6
2y = -12
y = -12/2
y = -6
5(-2) - 3y = 8
-10 - 3y = 8
-3y = 18
y = 18/-3
y = -6
Your answer is (-2,-6).
Answers
20
Step-by-step explanation:
is what i got i think its wrong tho
Hope it help u
A = 19 units2
A = 17 units2
A = 30 units
A = 18 units2
Answers
9514 1404 393
Answer:
(d) A = 18 units²
Step-by-step explanation:
Line CE can be considered to be the base of two triangles. It is 6 units long. The upper triangle is 4 units high, and the lower triangle is 2 units high. The formula for the area of a triangle is ...
A = 1/2bh
Then the sum of the areas of the two triangles is ...
A = 1/2(6)(4) +1/2(6)(2) = 12 +6 = 18
The area of the polygon is 18 square units.
Answer: A=18units2
Step-by-step explanation:
I said 17 before and got it wrong lol.
Next time I said 18 cuz it seemed like the closest when I estimated what it looked like the units it covered, so it's right:)
hope it helps!