Using the table, determine the expression that represents the value of each term as a function of its position, then find the value of the fifteenth term in the sequence.
Here we’re asked to do two things: determine the expression that represents the value of each term as a function, and then two find the value of the fifteenth term in the sequence.
Let’s start with determining the expression. I’ll want to look closely at how the value of terms are changing. From the first position to the second position, we go from four to nine; we’ve added five.
Is that the same pattern that happens when we go from position three to position four? It is; nine plus five equals 14.
And when we move from position four to position five, we move from 14 to 19, which is also adding five. That should indicate to us that whatever position we are in, we are multiplying that position by five; five times 𝑛.
Let’s test this out. I’m testing out position two. So I think that we’re multiplying five times our position. Five times two is ten, but the value of term two equals four.
So something else is going on here. How would we get from ten to four? We could subtract six, then we have five times two minus six equals four.
Perhaps our equation, our expression, is five 𝑛 minus six. Let’s check this expression with position three. If we’ve found the correct expression, we should have five times three minus six equals nine. Five times three plus 15; 15 minus six equals nine.
We’ll go ahead and check one more position just to make sure. Let’s check position five. Now we’ll need to know does five times five minus six equal 19. 25 minus six equals 19.
We’ve completed the first part; we’ve determined our expression five 𝑛 minus six, but the second part is asking us to find the value of the fifteenth term in the sequence. So we use our expression, five 𝑛 minus six, and we plug in the 𝑛 value of 15 because we’re looking for the fifteenth term.
Five times 15 equals 75, and 75 minus six equals 69.
Here are the two things this problem was looking for: 1) the expression five 𝑛 minus six, and 2) the value of the fifteenth term, which is 69.