2002 AMC 10B Problems/Problem 19
Contents
Problem
Suppose that is an arithmetic sequence with What is the value of
Solution 1
Adding the two given equations together gives
.
Now, let the common difference be . Notice that , so we merely need to find to get the answer. The formula for an arithmetic sum is
,
where is the first term, is the number of terms, and is the common difference. Now we use this formula to find a closed form for the first given equation and the sum of the given equations. For the first equation, we have . Therefore, we have
,
or
. *(1)
For the sum of the equations (shown at the beginning of the solution) we have , so
or
*(2)
Now we have a system of equations in terms of and . Subtracting (1) from (2) eliminates , yielding , and .
Solution 2
Subtracting the 2 given equations yields
Now express each a_n in terms of first term a_1 and common difference x between consecutive terms
Simplifying and canceling a_1 and x terms gives
See Also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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