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Find the given questions​

Find the given questions​-example-1

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9514 1404 393

Answer:

  • next term: 15
  • an = n(n+1)/2

Explanation:

Each term is the previous term with the current term number added. The 5th term will be 5 + (4th term) = 5 + 10 = 15.

The next term is 15.

_____

The sequence of triangular numbers is given by the explicit formula ...

an = n(n+1)/2

__

Finding the Formula

The sequence of first differences is ...

3 -1 = 2

6 -3 = 3

10 -6 = 4

And the sequence of second differences is ...

3 -2 = 1

4 -3 = 1

These are constant, indicating the sequence is described by a polynomial function of 2nd degree.

There are several ways to find the coefficients of the quadratic function. One is to solve for them using a system of equations.

n = 1

a1 = 1 = a(1)^2 +b(1) + c

n = 2

a2 = 3 = a(2)^2 + b(2) +c

n = 3

a3 = 6 = a(3)^2 +b(3) +c

To solve, we can subtract each equation from the next:

a2 -a1 = (3) -(1) = a(4 -1) +b(2 -1) +c(1 -1)

2 = 3a +b

and

a3 -a2 = (6) -(3) = a(9 -4) +b(3 -2) +c(1 -1)

3 = 5a +b

Now we can subtract the first of these equation from the second:

(3) -(2) = a(5 -3) +b(1 -1) ⇒ 1 = 2a ⇒ a = 1/2

Then ...

2 = 3(1/2) +b

1/2 = b

1 = a + b + c ⇒ 1 = 1/2 + 1/2 + c ⇒ c = 0

So, the quadratic function describing the sequence of triangular numbers is ...

an = (1/2)(n^2 +n)

an = n(n+1)/2

User Jonathan Lisic
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