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\(k, a_2, a_3\) and \(k, b_2, b_3\) are both nonconstant geometric sequences with different common ratios. We have \(a_3 -b_3 =3(a_2 -b_2)\) . Find the sum of the common ratios of the two sequences.

We can rewrite these sequences as \(k, ak, a^2k\) and \(k, bk, b^2k\) , since they are geometric sequences. 

We now have:

\(a^2k-b^2k = 3(ak-bk)\), which we can factor to become:

\(k(a^2-b^2)=3k(a-b)\).

We can rewrite:

\(k(a+b)(a-b)=3k(a-b)\)

Dividing by \(k(a-b)\) on both sides, we can get \(a+b=3\)

Since \(a\) is the common ratio of the first sequence, and \(b\) is the common ratio of the second, our answer is simply 3 .

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The last digit is 4. There’s a pattern: 4,6.... so just divide 35849 by 2. There’s a remainder of 1. So the answer is \(4\)

Last digits:

4^1 is 4

4^2 is 6

4^3 is 4

4^4 is 6...

35849 is odd, so it's 4.

The vertex of this parabola is (-1, 1). When you shift it 1 unit to the right and 5 units down, the vertex is now (-1+1, 1-5), which is (0, -4). So now our parabola equation is y=-(x)^2-4

Rotating it 180 degrees means flipping it upside down (so if it was facing downwards, now it's facing upwards, and vice versa). We know our parabola is now facing downward because its there's a negative sign in the original equation. 

If its now facing upward, then its positive. y=x^2-4 

Zeroes are when y=0. x^2-4=0

x^2 = 4

x = +2, -2

Since b≥a, we know that b=2, and a=-2. 

So b-a=2-(-2)=4

Use the Law of Cosines to find the angles:

Obtuse scalene triangle.

Sides: a = 125.1   b = 73.7   c = 80.6

Area: T = 2820.592
Perimeter: p = 279.4
Semiperimeter: s = 139.7

Angle ∠ A = α = 108.257° = 108°15'26″ = 1.889 rad
Angle ∠ B = β = 34.019° = 34°1'9″ = 0.594 rad
Angle ∠ C = γ = 37.723° = 37°43'24″ = 0.658 rad

n = 5, 12, 14, 15, 16
 

Thank You so much!!!

A circle equation is \((x-h)^2+(y-k)^2=r^2\), where (h, k) is the center and r is the radius.

We complete the square for x and y to get it in circle form.

Add 25 and 9 to both sides to complete the square.  \(x^2+10x+25+y^2+6y+9-k=25+9\)

Now we can write the squares. We also k to both sides. \((x+5)^2+(y+3)^2=34+k\)

It's in circle equation now!

Since the radius must be 6, 34+k=36. 

k=2

The name of the relationship between the circumference and the diameter is called  pi   

37037 , 37074 , 37148 , 37185 , 37259 , 37296 ,  37370 , 37407 , 37481 , 37518 ,  37592 , 37629 ,  37703 , 37740 ,  37814 , 37851 ,  37925 , 37962  = 18 such numbers 

Yes, I actually got it from alcums, and I also fixed it, sorry.

ABCDE is a pentagon with the sum of interior angles of 540º.

∠ABC+∠CDE = 270º    

I seriously don't understand why so many people like to copy questions with latex...

where do they come from???

Begin:o=0;e=0;listforeach(m, d=((1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 10, 12, 12, 12, 12, 14, 14, 15, 15, 16, 18, 18, 18, 18, 20, 20, 20, 20, 21, 21, 24, 24, 25, 27, 27, 28, 28, 30, 30, 30, 30, 35, 35, 36, 36, 36, 40, 40, 42, 42, 45, 45, 49, 50, 50, 54, 54, 60, 60, 63, 63, 70, 70, 81, 90, 90, 100);
if(m%2==0,e=e+1, o=o+1);print"Even Num =",e; print"Odd Num   =", o

OUTPUT:

Even products = 56
Odd products   = 25
Probability =56 / 81