How can I prove that the statement is true for every positive integer #n#?

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1 Answer
George C.
Feb 8, 2018

See explanation...

Explanation:

Proposition

Let #P(n)# be the proposition:

#8+16+24+...+8n = 4n(n+1)#

Base case

#P(1)# is true, since:

#8 = 4(color(blue)(1))((color(blue)(1)+1)#

Induction step

Suppose #P(n)# is true for some positive integer #n#:

#8+16+24+...+8n#

Then:

#8+16+24+...+8n+8(n+1) = 4n(n+1)+8(n+1)#

#color(white)(8+16+24+...+8n+8(n+1)) = (4n+8)(n+1)#

#color(white)(8+16+24+...+8n+8(n+1)) = 4(n+2)(n+1)#

#color(white)(8+16+24+...+8n+8(n+1)) = 4(color(blue)(n+1))((color(blue)(n+1))+1)#

So #P(n+1)# holds.

That is #P(n) => P(n+1)# for all positive integers #n#

Conclusion

Having shown #P(1)# and #P(n) => P(n+1)# for all positive integers #n#, we can deduce that #P(n)# for all positive integers #n# by induction.

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