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Make an inductive conclusion about the nth term in following pattern,...

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kasane | Student, College Freshman | Honors

Posted July 12, 2012 at 3:36 PM via web

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Make an inductive conclusion about the nth term in following pattern, `1+2=(2*3)/2,1+2+3=(3*4)/2,1+2+3+4=(4*5)/2,...`

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted July 12, 2012 at 4:27 PM (Answer #1)

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Given the following pattern:

`1+2=(2*3)/2,1+2+3=(3*4)/2,1+2+3+4=(4*5)/2,...`

Find an expression for the `n^(th)` term:

We are adding the natural numbers from 1 to `n` ; the sum is seen to be equal to the product of `n` and `n+1` all divided by 2.

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The inductive hypothesis is :

`1+2+3+***+n=(n(n+1))/2`  or `sum_1^n n=(n(n+1))/2`

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This sums form a well-known sequence called the triangular numbers.

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