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The sequence 9, 18, 27, 36, 45, 54,.. consists of successive multiples of 9. This sequence is then altered by multiplying every other term by (-1), starting with the first term, to produce the new sequence- 9, 18, -27, 36, 45, 54.... If the sum of the first n terms of this new sequence is 180, determine n.

 Nov 30, 2023
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here are the steps to solve the problem:

 

1. Find the formula for the terms of the new sequence.

The terms of the new sequence are alternately equal to 9, 18, −27, 36, and so on. This can be written as an​=(−1)n(9+9(n−1)).

 

2. Find the sum of the first n terms of the new sequence.

The sum of the first n terms of the new sequence is given by 21​∑i=1n​ai​=21​∑i=1n​(−1)n(9+9(n−1)). This can be simplified to 21​∑i=1n​(−1)i(9+9(i−1)).

 

3. Set the sum of the first n terms equal to 108 and solve for n.

We have 21​∑i=1n​(−1)i(9+9(i−1))=108. Multiplying both sides by 2, we get ∑i=1n​(−1)i(9+9(i−1))=216.

 

We can evaluate the sum on the left-hand side by pairing up the terms:

\begin{align*} \sum_{i=1}^n (-1)^i (9 + 9(i-1)) &= (9 - 18) + (27 - 36) + (45 - 54) + \dots + (a_{n-1} - a_n) \ &= 9 + 9 + 9 + \dots + 9 \ &= 9n. \end{align*}

 

Therefore, 9n=216, so n=24.

 Nov 30, 2023

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