Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
1970000-530000/(1+r)^36-56000*(1-1/(1+r)^36)/r
How do you 1970000-530000/(1+r)^36-56000*(1-1/(1+r)^36)/r in partial fractions?
The solution
You have entered
[src]
/ 1 \
56000*|1 - ---------|
| 36|
530000 \ (1 + r) /
1970000 - --------- - ---------------------
36 r
(1 + r)
$$\left(1970000 - \frac{530000}{\left(r + 1\right)^{36}}\right) - \frac{56000 \left(1 - \frac{1}{\left(r + 1\right)^{36}}\right)}{r}$$
1970000 - 530000/(1 + r)^36 - 56000*(1 - 1/(1 + r)^36)/r
General simplification
[src]
530000 56000 56000
1970000 - --------- - ----- + -----------
36 r 36
(1 + r) r*(1 + r)
$$1970000 - \frac{530000}{\left(r + 1\right)^{36}} - \frac{56000}{r} + \frac{56000}{r \left(r + 1\right)^{36}}$$
1970000 - 530000/(1 + r)^36 - 56000/r + 56000/(r*(1 + r)^36)
Fraction decomposition
[src]
1970000 - 586000/(1 + r)^36 - 56000/(1 + r) - 56000/(1 + r)^35 - 56000/(1 + r)^34 - 56000/(1 + r)^33 - 56000/(1 + r)^32 - 56000/(1 + r)^31 - 56000/(1 + r)^30 - 56000/(1 + r)^29 - 56000/(1 + r)^28 - 56000/(1 + r)^27 - 56000/(1 + r)^26 - 56000/(1 + r)^25 - 56000/(1 + r)^24 - 56000/(1 + r)^23 - 56000/(1 + r)^22 - 56000/(1 + r)^21 - 56000/(1 + r)^20 - 56000/(1 + r)^19 - 56000/(1 + r)^18 - 56000/(1 + r)^17 - 56000/(1 + r)^16 - 56000/(1 + r)^15 - 56000/(1 + r)^14 - 56000/(1 + r)^13 - 56000/(1 + r)^12 - 56000/(1 + r)^11 - 56000/(1 + r)^10 - 56000/(1 + r)^9 - 56000/(1 + r)^8 - 56000/(1 + r)^7 - 56000/(1 + r)^6 - 56000/(1 + r)^5 - 56000/(1 + r)^4 - 56000/(1 + r)^3 - 56000/(1 + r)^2
$$1970000 - \frac{56000}{r + 1} - \frac{56000}{\left(r + 1\right)^{2}} - \frac{56000}{\left(r + 1\right)^{3}} - \frac{56000}{\left(r + 1\right)^{4}} - \frac{56000}{\left(r + 1\right)^{5}} - \frac{56000}{\left(r + 1\right)^{6}} - \frac{56000}{\left(r + 1\right)^{7}} - \frac{56000}{\left(r + 1\right)^{8}} - \frac{56000}{\left(r + 1\right)^{9}} - \frac{56000}{\left(r + 1\right)^{10}} - \frac{56000}{\left(r + 1\right)^{11}} - \frac{56000}{\left(r + 1\right)^{12}} - \frac{56000}{\left(r + 1\right)^{13}} - \frac{56000}{\left(r + 1\right)^{14}} - \frac{56000}{\left(r + 1\right)^{15}} - \frac{56000}{\left(r + 1\right)^{16}} - \frac{56000}{\left(r + 1\right)^{17}} - \frac{56000}{\left(r + 1\right)^{18}} - \frac{56000}{\left(r + 1\right)^{19}} - \frac{56000}{\left(r + 1\right)^{20}} - \frac{56000}{\left(r + 1\right)^{21}} - \frac{56000}{\left(r + 1\right)^{22}} - \frac{56000}{\left(r + 1\right)^{23}} - \frac{56000}{\left(r + 1\right)^{24}} - \frac{56000}{\left(r + 1\right)^{25}} - \frac{56000}{\left(r + 1\right)^{26}} - \frac{56000}{\left(r + 1\right)^{27}} - \frac{56000}{\left(r + 1\right)^{28}} - \frac{56000}{\left(r + 1\right)^{29}} - \frac{56000}{\left(r + 1\right)^{30}} - \frac{56000}{\left(r + 1\right)^{31}} - \frac{56000}{\left(r + 1\right)^{32}} - \frac{56000}{\left(r + 1\right)^{33}} - \frac{56000}{\left(r + 1\right)^{34}} - \frac{56000}{\left(r + 1\right)^{35}} - \frac{586000}{\left(r + 1\right)^{36}}$$
586000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000 56000
1970000 - --------- - ----- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - --------- - -------- - -------- - -------- - -------- - -------- - -------- - -------- - --------
36 1 + r 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2
(1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r) (1 + r)
Numerical answer
[src]
1970000.0 - 530000.0/(1.0 + r)^36 - (56000.0 - 56000.0/(1.0 + r)^36)/r
1970000.0 - 530000.0/(1.0 + r)^36 - (56000.0 - 56000.0/(1.0 + r)^36)/r
Assemble expression
[src]
56000
56000 - ---------
36
530000 (1 + r)
1970000 - --------- - -----------------
36 r
(1 + r)
$$1970000 - \frac{530000}{\left(r + 1\right)^{36}} - \frac{56000 - \frac{56000}{\left(r + 1\right)^{36}}}{r}$$
1970000 - 530000/(1 + r)^36 - (56000 - 56000/(1 + r)^36)/r
Rational denominator
[src]
36 / 36\ 36 / 36\
(1 + r) *\56000 - 56000*(1 + r) / + 10000*r*(1 + r) *\-53 + 197*(1 + r) /
-----------------------------------------------------------------------------
72
r*(1 + r)
$$\frac{10000 r \left(r + 1\right)^{36} \left(197 \left(r + 1\right)^{36} - 53\right) + \left(56000 - 56000 \left(r + 1\right)^{36}\right) \left(r + 1\right)^{36}}{r \left(r + 1\right)^{72}}$$
((1 + r)^36*(56000 - 56000*(1 + r)^36) + 10000*r*(1 + r)^36*(-53 + 197*(1 + r)^36))/(r*(1 + r)^72)
Combinatorics
[src]
/ 36 35 2 34 3 33 4 32 5 31 6 30 7 29 8 28 9 27 10 26 11 25 12 24 13 23 14 22 15 21 16 20 17 19 18\
2000*\-288 + 985*r + 17820*r + 35432*r + 420630*r + 619542*r + 5383560*r + 7015260*r + 47465649*r + 57821505*r + 316798944*r + 369687780*r + 1684840080*r + 1908019344*r + 7375175280*r + 8167926624*r + 27170423060*r + 29572699860*r + 85613898832*r + 91883841280*r + 233551504872*r + 247738041320*r + 556746240960*r + 584675984592*r + 1168200425700*r + 1216079986212*r + 2169831434400*r + 2241080780400*r + 3583451470320*r + 3674650633200*r + 5279763602520*r + 5378087700000*r + 6957524123550*r + 7042352756670*r + 8214430362600*r + 8263913731920*r + 8698278365700*r /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
36
(1 + r)
$$\frac{2000 \left(985 r^{36} + 35432 r^{35} + 619542 r^{34} + 7015260 r^{33} + 57821505 r^{32} + 369687780 r^{31} + 1908019344 r^{30} + 8167926624 r^{29} + 29572699860 r^{28} + 91883841280 r^{27} + 247738041320 r^{26} + 584675984592 r^{25} + 1216079986212 r^{24} + 2241080780400 r^{23} + 3674650633200 r^{22} + 5378087700000 r^{21} + 7042352756670 r^{20} + 8263913731920 r^{19} + 8698278365700 r^{18} + 8214430362600 r^{17} + 6957524123550 r^{16} + 5279763602520 r^{15} + 3583451470320 r^{14} + 2169831434400 r^{13} + 1168200425700 r^{12} + 556746240960 r^{11} + 233551504872 r^{10} + 85613898832 r^{9} + 27170423060 r^{8} + 7375175280 r^{7} + 1684840080 r^{6} + 316798944 r^{5} + 47465649 r^{4} + 5383560 r^{3} + 420630 r^{2} + 17820 r - 288\right)}{\left(r + 1\right)^{36}}$$
2000*(-288 + 985*r^36 + 17820*r + 35432*r^35 + 420630*r^2 + 619542*r^34 + 5383560*r^3 + 7015260*r^33 + 47465649*r^4 + 57821505*r^32 + 316798944*r^5 + 369687780*r^31 + 1684840080*r^6 + 1908019344*r^30 + 7375175280*r^7 + 8167926624*r^29 + 27170423060*r^8 + 29572699860*r^28 + 85613898832*r^9 + 91883841280*r^27 + 233551504872*r^10 + 247738041320*r^26 + 556746240960*r^11 + 584675984592*r^25 + 1168200425700*r^12 + 1216079986212*r^24 + 2169831434400*r^13 + 2241080780400*r^23 + 3583451470320*r^14 + 3674650633200*r^22 + 5279763602520*r^15 + 5378087700000*r^21 + 6957524123550*r^16 + 7042352756670*r^20 + 8214430362600*r^17 + 8263913731920*r^19 + 8698278365700*r^18)/(1 + r)^36
Powers
[src]
56000
-56000 + ---------
36
530000 (1 + r)
1970000 - --------- + ------------------
36 r
(1 + r)
$$1970000 - \frac{530000}{\left(r + 1\right)^{36}} + \frac{-56000 + \frac{56000}{\left(r + 1\right)^{36}}}{r}$$
56000
56000 - ---------
36
530000 (1 + r)
1970000 - --------- - -----------------
36 r
(1 + r)
$$1970000 - \frac{530000}{\left(r + 1\right)^{36}} - \frac{56000 - \frac{56000}{\left(r + 1\right)^{36}}}{r}$$
1970000 - 530000/(1 + r)^36 - (56000 - 56000/(1 + r)^36)/r
Trigonometric part
[src]
56000
56000 - ---------
36
530000 (1 + r)
1970000 - --------- - -----------------
36 r
(1 + r)
$$1970000 - \frac{530000}{\left(r + 1\right)^{36}} - \frac{56000 - \frac{56000}{\left(r + 1\right)^{36}}}{r}$$
1970000 - 530000/(1 + r)^36 - (56000 - 56000/(1 + r)^36)/r
Combining rational expressions
[src]
/ 36 / 36\\
2000*\28 - 28*(1 + r) + 5*r*\-53 + 197*(1 + r) //
----------------------------------------------------
36
r*(1 + r)
$$\frac{2000 \left(5 r \left(197 \left(r + 1\right)^{36} - 53\right) - 28 \left(r + 1\right)^{36} + 28\right)}{r \left(r + 1\right)^{36}}$$
2000*(28 - 28*(1 + r)^36 + 5*r*(-53 + 197*(1 + r)^36))/(r*(1 + r)^36)
Common denominator
[src]
35 34 33 2 32 3 31 4 30 5 29 6 28 7 27 8 26 9 25 10 24 11 23 12 22 13 21 14 20 15 19 16 18 17
2546000 + 56000*r + 2016000*r + 35280000*r + 35280000*r + 399840000*r + 399840000*r + 3298680000*r + 3298680000*r + 21111552000*r + 21111552000*r + 109076352000*r + 109076352000*r + 467470080000*r + 467470080000*r + 1694579040000*r + 1694579040000*r + 5272023680000*r + 5272023680000*r + 14234463936000*r + 14234463936000*r + 33645096576000*r + 33645096576000*r + 70093951200000*r + 70093951200000*r + 129404217600000*r + 129404217600000*r + 212592643200000*r + 212592643200000*r + 311802543360000*r + 311802543360000*r + 409240838160000*r + 409240838160000*r + 481459809600000*r + 481459809600000*r + 508207576800000*r
1970000 - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
36 35 2 34 3 33 4 32 5 31 6 30 7 29 8 28 9 27 10 26 11 25 12 24 13 23 14 22 15 21 16 20 17 19 18
1 + r + 36*r + 36*r + 630*r + 630*r + 7140*r + 7140*r + 58905*r + 58905*r + 376992*r + 376992*r + 1947792*r + 1947792*r + 8347680*r + 8347680*r + 30260340*r + 30260340*r + 94143280*r + 94143280*r + 254186856*r + 254186856*r + 600805296*r + 600805296*r + 1251677700*r + 1251677700*r + 2310789600*r + 2310789600*r + 3796297200*r + 3796297200*r + 5567902560*r + 5567902560*r + 7307872110*r + 7307872110*r + 8597496600*r + 8597496600*r + 9075135300*r
$$- \frac{56000 r^{35} + 2016000 r^{34} + 35280000 r^{33} + 399840000 r^{32} + 3298680000 r^{31} + 21111552000 r^{30} + 109076352000 r^{29} + 467470080000 r^{28} + 1694579040000 r^{27} + 5272023680000 r^{26} + 14234463936000 r^{25} + 33645096576000 r^{24} + 70093951200000 r^{23} + 129404217600000 r^{22} + 212592643200000 r^{21} + 311802543360000 r^{20} + 409240838160000 r^{19} + 481459809600000 r^{18} + 508207576800000 r^{17} + 481459809600000 r^{16} + 409240838160000 r^{15} + 311802543360000 r^{14} + 212592643200000 r^{13} + 129404217600000 r^{12} + 70093951200000 r^{11} + 33645096576000 r^{10} + 14234463936000 r^{9} + 5272023680000 r^{8} + 1694579040000 r^{7} + 467470080000 r^{6} + 109076352000 r^{5} + 21111552000 r^{4} + 3298680000 r^{3} + 399840000 r^{2} + 35280000 r + 2546000}{r^{36} + 36 r^{35} + 630 r^{34} + 7140 r^{33} + 58905 r^{32} + 376992 r^{31} + 1947792 r^{30} + 8347680 r^{29} + 30260340 r^{28} + 94143280 r^{27} + 254186856 r^{26} + 600805296 r^{25} + 1251677700 r^{24} + 2310789600 r^{23} + 3796297200 r^{22} + 5567902560 r^{21} + 7307872110 r^{20} + 8597496600 r^{19} + 9075135300 r^{18} + 8597496600 r^{17} + 7307872110 r^{16} + 5567902560 r^{15} + 3796297200 r^{14} + 2310789600 r^{13} + 1251677700 r^{12} + 600805296 r^{11} + 254186856 r^{10} + 94143280 r^{9} + 30260340 r^{8} + 8347680 r^{7} + 1947792 r^{6} + 376992 r^{5} + 58905 r^{4} + 7140 r^{3} + 630 r^{2} + 36 r + 1} + 1970000$$
1970000 - (2546000 + 56000*r^35 + 2016000*r^34 + 35280000*r + 35280000*r^33 + 399840000*r^2 + 399840000*r^32 + 3298680000*r^3 + 3298680000*r^31 + 21111552000*r^4 + 21111552000*r^30 + 109076352000*r^5 + 109076352000*r^29 + 467470080000*r^6 + 467470080000*r^28 + 1694579040000*r^7 + 1694579040000*r^27 + 5272023680000*r^8 + 5272023680000*r^26 + 14234463936000*r^9 + 14234463936000*r^25 + 33645096576000*r^10 + 33645096576000*r^24 + 70093951200000*r^11 + 70093951200000*r^23 + 129404217600000*r^12 + 129404217600000*r^22 + 212592643200000*r^13 + 212592643200000*r^21 + 311802543360000*r^14 + 311802543360000*r^20 + 409240838160000*r^15 + 409240838160000*r^19 + 481459809600000*r^16 + 481459809600000*r^18 + 508207576800000*r^17)/(1 + r^36 + 36*r + 36*r^35 + 630*r^2 + 630*r^34 + 7140*r^3 + 7140*r^33 + 58905*r^4 + 58905*r^32 + 376992*r^5 + 376992*r^31 + 1947792*r^6 + 1947792*r^30 + 8347680*r^7 + 8347680*r^29 + 30260340*r^8 + 30260340*r^28 + 94143280*r^9 + 94143280*r^27 + 254186856*r^10 + 254186856*r^26 + 600805296*r^11 + 600805296*r^25 + 1251677700*r^12 + 1251677700*r^24 + 2310789600*r^13 + 2310789600*r^23 + 3796297200*r^14 + 3796297200*r^22 + 5567902560*r^15 + 5567902560*r^21 + 7307872110*r^16 + 7307872110*r^20 + 8597496600*r^17 + 8597496600*r^19 + 9075135300*r^18)