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[tex]U(n + 1) = kU(n) + r[/tex]
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[tex]n = 1[/tex]
[tex]U(1 + 1) = kU(1) + r[/tex]
[tex]U(2) = k \times U(1) + r[/tex]
[tex]188 = k \times (37) + r[/tex]
[tex]37k + r = 188 \: \: \: (( \alpha ))[/tex]_________________________________
[tex]n = 2[/tex]
[tex]U(2 + 1) = kU(2) + r[/tex]
[tex]U(3) = k \times U(2) + r[/tex]
[tex]943 = k \times (188) + r[/tex]
[tex]188k + r = 943 \: \: \: (( \beta ))[/tex]
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[tex]188k + r = 943 \: \: \: (( \beta ))[/tex]
[tex]37k + r = 188 \: \: \: \: \: (( \alpha ))[/tex]
[tex](( \beta )) - (( \alpha )) = [/tex]
[tex]188k - 37k + r - r = 943 - 188 \\ [/tex]
[tex]151k = 755[/tex]
Divide sides by 151
[tex] \frac{151k}{151} = \frac{755}{151} \\ [/tex]
[tex]k = 5[/tex]
Put the value of k in one of the equations to find the r :
[tex]37k + r = 188[/tex]
[tex]37 \times (5) + r = 188[/tex]
[tex]185 + r = 188[/tex]
[tex]r + 185 = 188[/tex]
Subtract sides 185
[tex]r + 185 - 185 = 188 - 185[/tex]
[tex]r = 3[/tex]
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