Question

Find a pattern in the set of equations, and use inductive reasoning to predict the next equation. Evaluate both sides of your new equation to check the results. Use your pattern to determine the 4th and the 8th equations. Enter equations using the + and power key

Answer 1

(a)The 4th Pattern is:

13+23+33+43+53=152

(b)The 8th Pattern is:

13+23+33+43+53+63+73+83+93=452

Explanation:

Given the set of equations:

13 + 23 = 32

13 + 23 + 33 = 62

13 + 23 + 33 + 43 = 102

On observation, the right-hand side in the set of equations above is represented using the function:

[TeX] U_{k-1}=2+\sum_{n=1}^{k} 10n [/TeX]

For the first equation, k=2

[TeX] U_{1}=U_{2-1}=2+\sum_{n=1}^{2} 10n =2+10+20=32[/TeX]

For the second equation, k=3

[TeX] U_{2}=U_{3-1}=2+\sum_{n=1}^{3} 10n =2+10+20+30=62[/TeX]

For the third equation, k=4

[TeX] U_{3}=U_{4-1}=2+\sum_{n=1}^{4} 10n =2+10+20+30+40=102[/TeX]

We are required to determine the 4th and 8th equations.

For the fourth equation, k=5

[TeX] U_{4}=U_{5-1}=2+\sum_{n=1}^{5} 10n =2+10+20+30+40+50=152[/TeX]

Therefore, the 4th equation is:

13+23+33+43+53=152

For the eight equation, k=9

[TeX] U_{8}=U_{9-1}=2+\sum_{n=1}^{9} 10n =2+10+20+30+40+50+60+70+80+90=452[/TeX]

Therefore, the 8th equation is:

13+23+33+43+53+63+73+83+93=452