+4 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.
+1725 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=1nai=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.
