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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

Let sk be a sequence such that:

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and we know that:

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Calculate the value of:

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Solution 1:

This solution does not use much knowledge and requires only work.

The relation:

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can be used to determine the value of s9:

equation

and can be solved for sn to relate its value to the two terms that precede it:

equation

equation

and we can use this recurrence formula repeatedly to compute terms ahead:

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or this formula:

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to compute preceding terms.

Now we can compute:

equation

equation

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Solution 2:

This solution requires more knowledge and less work.

Observe that the relation:

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expresses a simple fact about arithmetic sequences: any term is the average of the two neighboring terms.

Now we know that the sequence is arithmetic and we can compute the common difference:

equation

and we can compute any term using the common difference and the term's 'distance' from the terms that we know:

equation

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Even easier, we notice that s6 is two terms to the left of s8, while s12 is two terms to the right of s10. Therefore, whatever we subtract from s8 to get s6, we add to s10 to get s12.

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which is an observation that makes the solution way simpler.