All Answers 358201 Answers

101 + 103 + 105 + ... + 995 + 997 + 999

This is an arithmetic sequence with     t₁ = 101     t_n = 999     d = 2

We can use the formula:  t_n  =  t₁ + (n - 1)d  to find the number of terms:

--->     999  =  101 + (n - 1)2     --->     n = 450

We can use the formula:  Sum  =  n(t₁ + t_n)/2  to find the sum of the terms:

--->     Sum  =  450(101 + 999)/2     --->     Sum  =  247 500

If the sum were:  (1 - 2 + 3) + (4 - 5 + 6) + (7 - 8 + 9) + ... + (1999 - 2000 + 2001) + (2002 - 2003 + 2004)

we could do this:       2         +       5        +        8        + ... +             2000               +            2003

and we would have an arithmetic series with:  t₁ = 2     t_n = 2003     d = 3.

We could then use this formula;  t_n  =  t₁ + (n - 1)d  to find the number of terms:

     --->     2003  =  2 + (n - 1)3     --->     2001  =  (n - 1)3     --->     667  =  n - 1     --->   n  =  668  

and use this formula:  Sum  =  n(t₁ + t_n)/2  to find the sum:

     --->     Sum  =  668(2 + 2003)/2     --->     Sum  =  669 670

However, we don't have  2004  in the original problem, so we'll have to subtract that:

     --->     669 670 - 2004  =  667 666

\(\sum_{50}^{499}\)  2n+1 =247500

Perhaps you misunderstood my answer left for you to finish

        \(\sqrt[7]{1152}\)  =   2\(\sqrt[7]{9}\)   

a * b * c     would be    2 *7*9

We break the inequality down into two inequalities: \(\frac{1}{3}t-5 . We solve for  \(\frac{1}{3}t-5  first.

\(\frac{1}{3}t-5 -\frac{9}{2}\)

Then we solve for \(t-2\le-3t+7\).

\(t-2\le-3t+7\\ = 4t\le9\\ = t\le\frac{9}{4}\)

Then t must be \(-\frac{9}{2} . In interval notation this is  \(\boxed{(-\frac{9}{2}, \frac{9}{4}]}\).

from first equation    9-x  = y      sub into 2nd equation

x(9-x) =10

-x^2+9x-10 = 0      quadratic formula shows x = 9/2 +- sqrt41 /2          if x =      1.29844     or   7.70156

                                                                                                                 y =       7.70156    or   1.29844

then    x^3  + y^3  =   459

b-36 and b-33 must add up to b so we have:

(b-36)+(b-33)=b

2b-69=b

b=69

∠QNO = ∠NQO = 25º

 The measure of Arc QPN = 180º - 2*25º

Sorry young person! I only summed up the first 6 terms. Here are all the 10 terms:

S = 10/2 * [2*(293/3) + (28/3) * 9], solve for S

S =4190 / 3 =1396 + 2/3

2 \(\sqrt[7]{9}\)   perhaps? 

g = 21º

h = 71º

i = 80º

j = 38º

k = 92º

3/2 * [2*F + 2D]=321,  [3F + 12D] =405, solve for D, F
F =293/3 - This is your first term
D =28/3 - This is the common difference.
Sum = 6/2 * [2*(293/3) + (28/3) * 5]
Sum = 726

First thing, a line connecting the two points does NOT pass through the origin....

   so use the distance formula TWICE    once for the distance from  -4,5   to   0,0

       and once for the distance from    5,-4   to  0,0      then add them together

fiirst one       d^2 = (-4)^2 + 5^2

                     d = sqrt 41

second one   d^2 = 5^2 + -4^2

                      d = sqrt41

d = 2 sqrt 41    from point a to origin then to point b

The smallest a =11

The 2nd smallest a =51

11  +  51 = 62

As follows:

A rhombus has 4 sides that are equal, so

12x-456 = 11x-458     subtract 11x from both sides

x - 456 = -458             add 456 to both sides

x = -2

I'm so sorry...

x = 10       multiply both sides of the equation by 10

10* x  = 10*10

10x = 100

Don't know much about "differntial equations", but go to WolframAlpha and enter your equation as you have it have written. After you enter it, put a comma "," at end and say "Find x in terms of t". It will give you an answer. Good luck.

x = # of 4.75 tea         110-x = # of 2.50 tea

4.75 x    +  2.50 (110-x) = 3.85 (110)      solve for x ...the amount of 4.75 tea    then the amount of 2.50 tea is 110-x

Hour hand moves   0.5 degree per minute

    at 9:00 it has moved  270 degrees

        in 20 more minutes it moves 10 degrees = 280 degrees

minute hand moves 6 degrees per minute     at 20 minute = 120degrees

360 - 280 + 120 =    200^o

3x-1<5x+8

Solve by subtracting both sides by 3x

-1<2x+8

Subtract both sides by 8

-9<2x

Divide by 2

x>-9/2

5x+8<2x+15

Subtract both sides by 2x and 8

3x<7

x<7/3

Interval notation: (-9/2, 7/3)

You are very welcome!

:P