Solve frac{250^{circ}}{(1+10.25%)^4}+2500/(1+10.25%)^8+2500/(1+10.25%)^12+2500/(1+10.25%)^16

$$\frac { 250 ^ { \circ } } { ( 1 + 10.25 \% ) ^ { 4 } } + \frac { 2500 } { ( 1 + 10.25 \% ) ^ { 8 } } + \frac { 2500 } { ( 1 + 10.25 \% ) ^ { 12 } } + \frac { 2500 } { ( 1 + 10.25 \% ) ^ { 16 } }$$

Evaluate

$$\frac{5350283738799810740356394196882438400000000000}{2046526777500669368329342638102622164679041}\approx 2614.323837645$$

Solution Steps

Expand \frac{10.25}{100} by multiplying both numerator and the denominator by 100.

$$\frac{250}{\left(1+\frac{1025}{10000}\right)^{4}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{8}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Reduce the fraction \frac{1025}{10000} to lowest terms by extracting and canceling out 25.

$$\frac{250}{\left(1+\frac{41}{400}\right)^{4}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{8}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Add 1 and \frac{41}{400} to get \frac{441}{400}.

$$\frac{250}{\left(\frac{441}{400}\right)^{4}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{8}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Calculate \frac{441}{400} to the power of 4 and get \frac{37822859361}{25600000000}.

$$\frac{250}{\frac{37822859361}{25600000000}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{8}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Divide 250 by \frac{37822859361}{25600000000} by multiplying 250 by the reciprocal of \frac{37822859361}{25600000000}.

$$250\times \frac{25600000000}{37822859361}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{8}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Multiply 250 and \frac{25600000000}{37822859361} to get \frac{6400000000000}{37822859361}.

$$\frac{6400000000000}{37822859361}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{8}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Expand \frac{10.25}{100} by multiplying both numerator and the denominator by 100.

$$\frac{6400000000000}{37822859361}+\frac{2500}{\left(1+\frac{1025}{10000}\right)^{8}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Reduce the fraction \frac{1025}{10000} to lowest terms by extracting and canceling out 25.

$$\frac{6400000000000}{37822859361}+\frac{2500}{\left(1+\frac{41}{400}\right)^{8}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Add 1 and \frac{41}{400} to get \frac{441}{400}.

$$\frac{6400000000000}{37822859361}+\frac{2500}{\left(\frac{441}{400}\right)^{8}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Calculate \frac{441}{400} to the power of 8 and get \frac{1430568690241985328321}{655360000000000000000}.

$$\frac{6400000000000}{37822859361}+\frac{2500}{\frac{1430568690241985328321}{655360000000000000000}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Divide 2500 by \frac{1430568690241985328321}{655360000000000000000} by multiplying 2500 by the reciprocal of \frac{1430568690241985328321}{655360000000000000000}.

$$\frac{6400000000000}{37822859361}+2500\times \frac{655360000000000000000}{1430568690241985328321}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Multiply 2500 and \frac{655360000000000000000}{1430568690241985328321} to get \frac{1638400000000000000000000}{1430568690241985328321}.

$$\frac{6400000000000}{37822859361}+\frac{1638400000000000000000000}{1430568690241985328321}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Add \frac{6400000000000}{37822859361} and \frac{1638400000000000000000000}{1430568690241985328321} to get \frac{1880466299910400000000000}{1430568690241985328321}.

$$\frac{1880466299910400000000000}{1430568690241985328321}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Expand \frac{10.25}{100} by multiplying both numerator and the denominator by 100.

$$\frac{1880466299910400000000000}{1430568690241985328321}+\frac{2500}{\left(1+\frac{1025}{10000}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Reduce the fraction \frac{1025}{10000} to lowest terms by extracting and canceling out 25.

$$\frac{1880466299910400000000000}{1430568690241985328321}+\frac{2500}{\left(1+\frac{41}{400}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Add 1 and \frac{41}{400} to get \frac{441}{400}.

$$\frac{1880466299910400000000000}{1430568690241985328321}+\frac{2500}{\left(\frac{441}{400}\right)^{12}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Calculate \frac{441}{400} to the power of 12 and get \frac{54108198377272584130510593262881}{16777216000000000000000000000000}.

$$\frac{1880466299910400000000000}{1430568690241985328321}+\frac{2500}{\frac{54108198377272584130510593262881}{16777216000000000000000000000000}}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Divide 2500 by \frac{54108198377272584130510593262881}{16777216000000000000000000000000} by multiplying 2500 by the reciprocal of \frac{54108198377272584130510593262881}{16777216000000000000000000000000}.

$$\frac{1880466299910400000000000}{1430568690241985328321}+2500\times \frac{16777216000000000000000000000000}{54108198377272584130510593262881}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Multiply 2500 and \frac{16777216000000000000000000000000}{54108198377272584130510593262881} to get \frac{41943040000000000000000000000000000}{54108198377272584130510593262881}.

$$\frac{1880466299910400000000000}{1430568690241985328321}+\frac{41943040000000000000000000000000000}{54108198377272584130510593262881}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Add \frac{1880466299910400000000000}{1430568690241985328321} and \frac{41943040000000000000000000000000000}{54108198377272584130510593262881} to get \frac{113067652394611106101254400000000000}{54108198377272584130510593262881}.

$$\frac{113067652394611106101254400000000000}{54108198377272584130510593262881}+\frac{2500}{\left(1+\frac{10.25}{100}\right)^{16}}$$

Expand \frac{10.25}{100} by multiplying both numerator and the denominator by 100.

$$\frac{113067652394611106101254400000000000}{54108198377272584130510593262881}+\frac{2500}{\left(1+\frac{1025}{10000}\right)^{16}}$$

Reduce the fraction \frac{1025}{10000} to lowest terms by extracting and canceling out 25.

$$\frac{113067652394611106101254400000000000}{54108198377272584130510593262881}+\frac{2500}{\left(1+\frac{41}{400}\right)^{16}}$$

Add 1 and \frac{41}{400} to get \frac{441}{400}.

$$\frac{113067652394611106101254400000000000}{54108198377272584130510593262881}+\frac{2500}{\left(\frac{441}{400}\right)^{16}}$$

Calculate \frac{441}{400} to the power of 16 and get \frac{2046526777500669368329342638102622164679041}{429496729600000000000000000000000000000000}.

$$\frac{113067652394611106101254400000000000}{54108198377272584130510593262881}+\frac{2500}{\frac{2046526777500669368329342638102622164679041}{429496729600000000000000000000000000000000}}$$

Divide 2500 by \frac{2046526777500669368329342638102622164679041}{429496729600000000000000000000000000000000} by multiplying 2500 by the reciprocal of \frac{2046526777500669368329342638102622164679041}{429496729600000000000000000000000000000000}.

$$\frac{113067652394611106101254400000000000}{54108198377272584130510593262881}+2500\times \frac{429496729600000000000000000000000000000000}{2046526777500669368329342638102622164679041}$$

Multiply 2500 and \frac{429496729600000000000000000000000000000000}{2046526777500669368329342638102622164679041} to get \frac{1073741824000000000000000000000000000000000000}{2046526777500669368329342638102622164679041}.

$$\frac{113067652394611106101254400000000000}{54108198377272584130510593262881}+\frac{1073741824000000000000000000000000000000000000}{2046526777500669368329342638102622164679041}$$

Add \frac{113067652394611106101254400000000000}{54108198377272584130510593262881} and \frac{1073741824000000000000000000000000000000000000}{2046526777500669368329342638102622164679041} to get \frac{5350283738799810740356394196882438400000000000}{2046526777500669368329342638102622164679041}.

$$\frac{5350283738799810740356394196882438400000000000}{2046526777500669368329342638102622164679041}$$

Factor

$$\frac{47 \cdot 547 \cdot 3709 \cdot 18521 \cdot 233321477 \cdot 2028783359453 \cdot 2 ^ {17} \cdot 5 ^ {11}}{3 ^ {32} \cdot 7 ^ {32}} = 2614\frac{6.627424130607537 \times 10^{41}}{2.0465267775006692 \times 10^{42}} = 2614.3238376454915$$