Video Transcript
Find the values of 𝑙 and 𝑚, where 𝑙 is equal to 𝑚 minus 32 given the arithmetic sequence negative one, 𝑚, and so on, 𝑙, negative 41.
We know that in any arithmetic sequence the difference between any two consecutive terms is always the same. This difference is known as the common difference and is denoted by the letter 𝑑. From the arithmetic sequence given, we can create two equations. Firstly, negative one plus 𝑑 is equal to 𝑚. Adding one to both sides of this equation gives us 𝑑 is equal to 𝑚 plus one. We will call this equation one. Using the last two terms of our sequence, we see that 𝑙 plus 𝑑 is equal to negative 41. Using equation one, we can substitute 𝑚 plus one for 𝑑. This means that 𝑙 plus 𝑚 plus one is equal to negative 41. We can then subtract one from both sides such that 𝑙 plus 𝑚 is equal to negative 42. We will call this equation two.
We were also told in the question that 𝑙 is equal to 𝑚 minus 32. Replacing 𝑙 with 𝑚 minus 32 in equation two gives us 𝑚 minus 32 plus 𝑚 is equal to negative 42. Collecting the like terms, we have two 𝑚 minus 32 on the left-hand side. We can then add 32 to both sides of this equation such that two 𝑚 is equal to negative 10. Finally, dividing both sides by two gives us 𝑚 is equal to negative five. We can now substitute this value of 𝑚 back into the equation 𝑙 is equal two 𝑚 minus 32. This gives us 𝑙 is equal to negative five minus 32. 𝑙 is therefore equal to negative 37.
The values of 𝑙 and 𝑚 that satisfy the arithmetic sequence such that 𝑙 is equal to 𝑚 minus 32 are negative 37 and negative five, respectively.
