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This essential trains for: AMC-8, AMC-10, Math Kangaroo 7-8, Math Kangaroo 9-10.

This is a nice method for students who are too young to be taught geometric sequences. Older students who are competitive will also benefit from the different angle that this method provides.

Take the sum of a geometric sequence:

equation

and notice that, by adding the fraction:

equation

we manage to shrink the sum by one term.

equation

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Not only that, but the last term is of the same form as the 'seed' term we added and can be used to recursively shrink the sum.

equation

Proceeding to shrink the sum as outlined above will eventually leave only one last term:

equation

We can solve for S:

equation

equation

Figure out what 'seed' term you have to add to the sum below in order to shrink it and compute its value:

equation