Please help, this is urgent

Question

What is the explicit formula for the arithmetic sequence defined by a_n =a_n-1 -3 if a₁ =4?

Answer 1

Hello Aaron!

If a1 =4, then a2 = 4-3 = 1

a3 = a2 - 3 = 4 - 3 -3 = 4-3*2 = -2

a4 = a3 - 3 = 4 - 3 -3*2 = 4 - 3*3 = 4 - 9 = -5

The formula seem to be: a_n = 4 -3*(n-1) . O get a more accurate answer, we need to test of it holds for other values, it will be clear later. Let's consider a_n+1.

By the question statement, a_n+1 = a_n - 3. If our formula is equivalent to this one, the it must be true (because if it's true for a term, it must hold true to the other). By our formula.

a_n+1 = 4 -3*n

a_n = 4 -3*(n-1)

a_n+1 - a_n = -3*n + 3*(n-1) = -3n + 3n -3 = -3

a_n+1 = a_n - 3

Therefore our formula is true.

a_n = 4 -3*(n-1)

Answer 2

'Explicit formula' means an expression which relates the value of the number, a_n in the Arithmetic sequence to it's position in the sequence, n.

Given

a_n = a_(n-1) - 3 --- 1

and

a₁ = 4 ---- 2

Use n=1 in eq 1

a₁ = a₀ - 3

From the given value of a₁ in eq 2, you can calculate a₀ = 7.

Now write the first few numbers in the series a₀, a₁, a_2....

n = 0, 1, 2, 3, 4...............

a_n = 7, 4, 1, -2, -5..............

Notice that the number value at a given position n can be obtained from the first number a0 in the series by subtracting 3, 'n' number of times. e.g.

for n =4, you can calculate

a₄ = 7 - 3*4

You can generalize this to get the explicit formula

a_n = 7 - 3*n

Answer 3

Based on the recursive formula, we know that each term (a_n) is the term before it (a_n-1) minus 3. Since the amount that is changing is constant, we know that this is a linear relationship, in the form of y = mx + b. Our y will be a_n, our x is n, and we need to find the slope m and y-intercept b.

Thus, it follows that for each time that n increases, a_n should decrease by 3. This means that our m value is -3:

a_n = -3n + b

We don't know yet, however, what the y-intercept b is. To find this, we'll plug in the other piece of information given to us: a₁ = 4. In this case, a_n is 4, and n is 1.

4 = -3(1) + b

We find that b is equal to 7, and so our full explicit formula for the arithmetic sequence is as follows:

a_n = -3n + 7

I hope my answer was able to clear things up for you; if not, don't hesitate to ask any questions!