How many terms must be added in an arithmetic sequence whose first term is 11 and whose common d

How many terms must be added in an arithmetic sequence whose...

Key Concepts

Arithmetic Series Sum Formula

The arithmetic series sum formula is used to calculate the sum of the first n terms of an arithmetic sequence. It is typically written as S = n/2 × (first term + last term) or S = n/2 × [2a + (n-1)d], where a is the first term and d is the common difference. This formula is essential for finding the number of terms needed to reach a particular sum.

Solving Quadratic Equations

When using the arithmetic series sum formula to solve for the number of terms, the resulting equation is often quadratic in nature. Solving quadratic equations, whether by factoring, completing the square, or using the quadratic formula, is an important skill that enables one to determine the possible values of n that satisfy the sum condition.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference allows for the prediction of subsequent terms and provides the foundation for various formulas, such as the formula for the n-th term and for the sum of the sequence.

Common Difference

The common difference is the fixed amount added to each term in an arithmetic sequence to obtain the next term. It is a critical component because it defines the progression of the sequence and is used in the formulas for the n-th term and the sum of the series.

Transcript

00:01 So in this problem, we're being asked how many terms have to be added in the arithmetic sequence where the first term is 11 and the common difference is 3, where the sum is 1092.

00:11 So because they're asking us how many terms, we're being asked to find n.

00:15 Well, we're told the first term is 11, so a sub 1 is 11.

00:19 The common difference is 3, so d is 3.

00:22 And our sum is 1092.

00:25 So we know s of n is 1092.

00:26 To do this, we're going to use the formula, s of n divided by 2, times the quantity of 2 times a sub 1, plus n minus 1 times d.

00:38 So let's substitute in the values that we know.

00:41 Well, we have 1 ,092 is equal to n divided by 2 times the quantity of 2 times 11 plus n minus 1 times 3.

00:53 So, we're going to have 1 ,092 equals n over 2, and inside our preemphasies, we have 2 times 11, which is 22.

01:01 And then if we distribute the 3, we're going to get 3n minus 3.

01:07 Well, 22 minus 3 is 19.

01:10 So if we're going to have 1092 is equal to n divided by 2 times the quantity of 19 plus 3n.

01:18 Now, what i'm going to do next is multiply both sides of my equation by 2, so that way i get rid of my fraction.

01:24 Well, 1092 times 2 is equal to 2 ,184.

01:32 And this is going to be equal to n times the quantity of 19 plus 3n.

01:38 Well, when i distribute the n, we're going to be left with 2 ,184 is equal to 19n plus 3n squared.

01:48 So what i can do is set this equal to 0 because i see that i have a quadratic equation here.

01:54 So when i do this, i'm going to end up with 3n squared plus 19n minus 2 ,184, and this is all going to be equal to 0.

02:08 So now what we want to try and do is factor this equation.

02:12 So we have to think of what might multiply to 2 ,184 that's somehow going to multiply to 19...

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