The Sequence:
1 , 1+2 , 1+2+3 , 1+2+3+4, ...
here, every term is an AP. finding the general formula for the sum of the elements of each term is:
Sₙ = [tex]\frac{x}{2}(2a + (x-1)d)[/tex] [where x = number of term, a = first term and d = common difference]
here, the first term is always 1 and so is the common difference.
Sₙ = [tex]\frac{x}{2}(2 +x-1)[/tex]
Sₙ = [tex]\frac{x}{2}(1 +x)[/tex] = [tex]\frac{1}{2}(x + x^{2})[/tex]
which is the formula for a general term in our series
now, we need to find the sum of the first n terms of this series
[tex]\displaystyle\sum_{x=1}^{n} [\frac{1}{2}(x + x^{2})][/tex]
[tex]\displaystyle\frac{1}{2} [\sum_{x=1}^{n} (x) + \sum_{x=1}^{n}(x^{2})][/tex]
in this formula, for the first term, it's just an AP from x = 1 to x = n
for the second term, we have a general formula [tex]\frac{n(n+1)(2n+1)}{6}[/tex]
[tex]\frac{1}{2}[\frac{n}{2}(2a + (n-1)d)+ \frac{n(n+1)(2n+1)}{6} ][/tex]
in this AP (first term), the first term and the common difference is 1 as well
[tex]\frac{1}{2}[\frac{n}{2}(2 + n-1)+ \frac{n(n+1)(2n+1)}{6} ][/tex]
[tex]\frac{1}{2}[\frac{n}{2}(n+1)+ \frac{n(n+1)(2n+1)}{6} ][/tex]
[tex][\frac{n}{4}(n+1)+ \frac{n(n+1)(2n+1)}{12} ][/tex]
[tex]\frac{n}{4}(n+1) [1+\frac{(2n+1)}{3} ][/tex]
[tex]\frac{n}{4}(n+1) [\frac{(3+2n+1)}{3} ][/tex]
[tex]\frac{n}{4}(n+1) [\frac{(2n+4)}{3} ][/tex]
[tex]\frac{n}{2}(n+1) [\frac{(n+2)}{3} ][/tex]
[tex]\frac{n(n+1)(n+2)}{6}[/tex]
which is the sum of n terms of the given sequence
what is a × 2. what is a(2)
Answer:
a × 2 means 'a' multiplied by 2
a(2) also means 'a' multiplied by 2
So both mean the same.
a × 2 means 2 times the value of variable "a".
a(2) represents a function "a" with input 2.
a × 2 means the value of variable "a" multiplied by 2. In mathematical terms, it is simply "2 times a" or "2 * a."
a(2) indicates a function notation, where "a" is a function and "2" is the input to that function.
It is not multiplication but rather a way of representing the value of the function "a" when the input is 2.
Hence, a × 2 means 2 times the value of variable "a".
a(2) represents a function "a" with input 2.
To learn more on Number system click:
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Seven less than the product of 8 and a number equals 4
The perimeter of an equilateral triangle is 15 inches more than the perimeter of a square, and the side of the triangle is 7 inches longer than the side of the square.
Find the side of the triangle (Hint: An equilateral triangle has three sides the same length.)
The side of the triangle is __in3. - in. - in2.
Solve the equation with all steps.
3/2x-2/3x=2
Answer:
[tex]x = \frac{5}{12} [/tex]
Step-by-step explanation:
[tex] \frac{3}{2x} - \frac{2}{3x} [/tex]
➡️ [tex] \frac{3}{2x} - \frac{2}{3x} = 2[/tex]
➡️ [tex] \frac{5}{6x} = 2[/tex]
➡️ [tex]5 = 12x = 12x = 5[/tex]
➡️ [tex]x = \frac{5}{12} [/tex]
Answer:
x = [tex]\frac{12}{5}[/tex]
Step-by-step explanation:
Given
[tex]\frac{3}{2}[/tex] x - [tex]\frac{2}{3}[/tex] x = 2 ( multiply through by 6 to clear the fractions )
9x - 4x = 12
5x = 12 ( divide both sides by 5 )
x = [tex]\frac{12}{5}[/tex]
3/5 of the T-shirts and a T-shirt shop are blue 5/8 of those T-shirts are on sale 1/3 of the blue T-shirts that are on sale or size medium what fraction of the shops T-shirts are blue T-shirts that are on sale and are size medium
Answer:
yes their size is medium
find m<YDC
please help!
Answer:
∠YDC measures 140°.
Step-by-step explanation:
We can consider using the Exterior Angle Theorem. According to the theorem, the exterior angle of a triangle is equal to the two opposite interior angles.
In other words:
[tex]\displaystyle m \angle YDC = m\angle DCB + m\angle CBD[/tex]
Substitute:
[tex]\displaystyle (15x + 5) = (80) + (6x+6)[/tex]
Solve for x. Combine like terms:
[tex]15x + 5 = 6x + 86[/tex]
Simplify:
[tex]9x = 81[/tex]
And divide. Hence, the value of x is:
[tex]x = 9[/tex]
∠YDC is given by:
[tex]\displaystyle m\angle YDC = 15x + 5[/tex]
Since we now know the value of x, substitute and evaluate:
[tex]\displaystyle \begin{aligned} m\angle YDC &= 15x + 5 \\ &= 15(9) + 5 \\ &= 135 + 5 \\ &= 140^\circ\end{aligned}[/tex]
In conclusion, ∠YDC measures 140°.
can anyone please help me solve it
11,12 (1)(2)(3),13 this is all
Answer:
answer of 11
Step-by-step explanation:
given A=30°
R.H.S=2sinA cosA
=2 * sin30° * cos30°
[tex] 2 \times \frac{1}{2} \times \frac{ \sqrt{3} }{2} [/tex]
[tex] = \frac{ \sqrt{3} }{2} [/tex]
=sin60°
sin 2*30°
=sin2A
L.H.S=R.H.S
A rectangular trampoline has an area of 204 squaſe feet. The length of the trampoline is 17 feet. What is the width?
Answer:
width = 12 ft
Step-by-step explanation:
A = w * l
w = a/ 1 = 204/ 17 = 12 ft
Alphonso was earning S1,860 per month
and then got a 12% raise. How much
will he make per month now?
Answer:
2,083.3
Step-by-step explanation:
can someone help me ?
Answer:
r = - 29
Step-by-step explanation:
Given
- 3r = 87 ( divide both sides by - 3 )
r = [tex]\frac{87}{-3}[/tex] = - 29
Answer:
here,we have-3r= 87now,r=87÷ -3
r= -29
Which number below comes first when the numbers are listed from least to greatest? Explain. 22 16 -3, 817, - 4.5 7 5 ... Which number comes first when the numbers are listed from least to greatest?
A. - 4.5 because it is the negative number with the greatest absolute value.
В. 16 because it is the greatest positive number. 5
C. V17 because it is the greatest positive number.
D. -3 because it is the negative number with the greatest absolute value. 22 hen click Check Answer
Answer: Choice A.
- 4.5 because it is the negative number with the greatest absolute value.
===================================================
Explanation:
Stuff to the left is smaller than stuff on the right for any number line.
For example, we can say 2 < 5 since 2 is to the left of 5.
Another example: -7 < 4
To find the smallest item of your given list, we need to look at the value that is furthest to the left. That value would be -4.5 and this value is negative with the largest absolute value (i.e. it is furthest away from 0).
Check out the number line diagram below. It shows where all of the values are placed. We can see that -4.5 is the left-most item.
i need helpp !! ill mark brainliest !
393/1000. i think, sorry. :c
3y+7<2-y help doing a test rn please help fast
Answer:
y<-5/4
Step-by-step explanation:
3y+7<2-y
-7 on both sides
3y<-5-y
+y on both sides
4y<-5
divide by 4 on both sides
y<-5/4
please help me with these. i need the steps and everything. i’m so lost
Answer:
Are you in geometry?
For question 5:
To solve for x:
Triangle to the right with the 72 degrees in one angle.
All triangles add up to 180.
Subtact 72 from 180 = 108
Now we need to find the two other angles of that triangle.
We know that the two given eqations for that triangle is 6x-2 and 11x-9 (remeber the property creating the area of intersection, two congruent angles, so the equation 11x-9 applys for both triangles.)
So for the triangle to the right;
(6x-2) + (11x-9) = 108
lets add like terms next
17x - 11 = 108
add 11 to 108
17x = 119
divide 17 from both sides
x = 7
Now we need to find y.
On triangle to the right we have 30degrees. Lets subtract 30 degrees from 180. This equals 150.
We can plug in 7 to the equation 11x-9 which give us 68.
We now know to angles of the triangle. Lets subtract 68 from 150.
We get 82.
The last angle is 82, Which means lets find y.
5y - 8 = 82
add 8 to both sides
5y = 90
divide by 5 to both sides
y = 18
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try number 6. on your own to see if you get it know. If you don't, comment to me regarding what you don't understand.
what is the common denominator of 4/5 and 9/4
hope this helps some
A town’s population increases at a rate of 2.3% every year. The current population is 7,500 people. Which equation models this scenario?
Answer:
7500 ( 1.023 ) ^(x)
Step-by-step explanation:
x is the number of years that go by
Find the equation of the line that contains the point (21,27) and is parallel to the x-axis.
Answer:
y - 27 = 0 (x - 21) or y = 27
Step-by-step explanation:
compare each number in the top row of the table with the number below it describing any pattern you see complete the table with a variable expression involving n.
1 2 3 4. ... n
1 16 81 256 ... ?
Answer:
The pattern is [tex]n^{4}[/tex].
Step-by-step explanation:
[tex]1^{4}[/tex] = 1, [tex]2^{4}[/tex] = 16, [tex]3^{4}[/tex] = 81, [tex]4^{4}[/tex] = 256
So the pattern is [tex]n^{4}[/tex].
? divided by 9> 700
like whats the answer
im confused help......
find the value :
of x
[tex]\\ \rm\longmapsto <AEC+<CED+x=180[/tex]
[tex]\\ \rm\longmapsto 47+90+x=180[/tex]
[tex]\\ \rm\longmapsto 137+x=180[/tex]
[tex]\\ \rm\longmapsto x=180-137[/tex]
[tex]\\ \rm\longmapsto x=43[/tex]
Help me with this math question plss
Answer:
C. 8/17
Step-by-step explanation:
PM = 16+14 = 30 cm
KP= LN = 16×tan45° = 16×1 = 16cm
by using pythagorean theorem, we find :
KM² = PM²+KP²
= 30²+16²
= 900+256
= 1156
=> KM =√1156 = 34cm
so, cos theta = KP/KM = 16/34 = 8/17
If you roll two dices, what is your probability not get 2 on both dice ？
Answer:
10/12 or 5/6
Step-by-step explanation:
So assuming each dice has 6 sides
6+6=12
12-2=10
10/12 or 5/6
Answer:
possible outcomes =1,1,2,2,3,3,4,4,5,5,6,6
probability that you don't get 2 on both dice is
10/12
=5/2
Help me
6 = 1/5 (4y + 10)
Step-by-step explanation:
open the bracket
6={(1/5×4y) + 1/5×10}
6=4/5y+2
6-2=4/5y
4=4/5y
divide both sides by 4/5
y=1/5
The answer you are looking for is y=5.
Solution/Explanation:
Writing out the problem,
6=1/5(4y+10)
Using the Distributive Property,
6=4/5y+2
Subtracting 2 from both sides of the equation,
6-2=4/5y
4=4/5y
Multiplying both sides by 5, to transform the fraction into a whole number,
20=4y
Finally, dividing both sides by 4,
So, therefore the final answer is y=5.
4/6 into a decimal then round to nearest hundredth full explanation
Answer:
0.67
Step-by-step explanation:
4/6 = 0.66666667
The 6 in the thousandth place is greater than 5, therefore we add 1 to the 6 that's in the hundredth place.
I hope this helps :)
what is Eight decreased by 3 times a number is 2
Answer:
8- 3x = 2 I think this is
Answer: The unknown number is 2.
Step-by-step explanation:
Let the number be x
8-3x = 2
or, -3x = -6
so, x = -6/-3
so, x = 2
A ball is thrown into the air with an upward velocity of 32 ft/s. It’s height h in feet after t seconds is given by the function h = -16t + 32t + 6 in how many seconds does the ball reach it maximum height? Round to the nearest hundredth if necessary
Answer:
gfxdg
Step-by-step explanation:
Statements Reasons 1. Given 1. Given 4 2. RS + ST = PS + SQ 2. Addition Property 3. RS + ST = RT; PS + SQ = PQ 3. 4. RS=ST;RQ=ST 4. Substitution Property
can someone help me
please??
Cho hình chóp có cạnh đáy là hình vuông cạnh bằng . Tam giác cân tại và mặt bên vuông góc với đáy, biết . Tính thể tích của khối chóp .
Answer:
Ah Here We Go Again :) After Five Years
Gear X has 12 cogs (teeth) and turns
clockwise at 30 revolutions per
minute. Gear Y has 24 cogs and gear
Z has 6 cogs
How fast and in which direction does
gear Z turn?