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I have three questions:

1. Let $a_1,$ $a_2,$ $a_3,$ $\dots,$ $a_{10},$ $a_{11},$ $a_{12}$ be an arithmetic sequence. If $a_1 + a_3 + a_5 + a_7 + a_9 + a_{11} = -2$ and $a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12} = 1$, then find $a_1$.

 

2. Find the sum of all positive integers less than $1000$ ending in $3$ or $4.$

 

3. When the same constant is added to the numbers $60,$ $100,$ and $165,$ a three-term geometric sequence arises. What is the common ratio of the resulting sequence?

 

Thank you in advance!

 Jan 12, 2024

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