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Let $a_1, a_2, a_3, \dots$ be a non-constant arithmetic sequence. (That is, not all the terms are the same.) Suppose that $a_4, a_7, a_{12}$ form a geometric sequence. If $a_7 = 30$, what is $a_1$?

 Feb 13, 2023
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In this case, we can use the formula for a geometric sequence to calculate the value of a1. The formula is an = a1 * r^(n-1), where r is the common ratio and a1 is the first term of the sequence.

 

Since a7 = 30, we can solve for a1 by rearranging the equation to a1 = a7/r^6. To calculate the common ratio, we can use the formula r = am/an, where m and n are two different terms in the sequence (in this case, a4 and a7).

 

Plugging in the values for a4, a7, and r gives us a1 = 30/3^6 = 10/243.

 Feb 13, 2023

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