Write a general formula for the arithmetic sequence given.(B) 12,56,76,32,…\frac12,\frac56,\frac76,\frac32,\ldots

a n = 1 3 n + 1 6 a_{n}=\frac13n+\frac16 a n = 3 1 n + 6 1

Write a general formula for the arithmetic sequence given. (B) 1 2 , 5 6 , 7 6 , 3 2 , … \frac12,\frac56,\frac76,\frac32,\ldots

we're asked to write a general formula for the arithmetic sequence given. The sequence we're given here is one half five six seven six three halves. Now we need two things in order to write our general formula for an arithmetic sequence. And that is our first term and our common difference. Now our first term here is given. It is one half based on our sequence. And we just need to then find our common difference, d. I want to do that by looking at two consecutive terms. And looking at these, I'm going to go ahead and choose two consecutive terms that have the same denominator so that I don't have to do anything extra here. So I'm going to pick these two middle terms, five sixths and seven sixths. So subtracting a three minuteus a two gives me sevensix minuteus fivesix, which is twosix or onethree. Now I can verify that this is indeed my common difference by just looking at what's happening between my other terms. So if I think of this one half as being threesix, so giving it a common denominator, if I take threesix and add twosix, that gives me my next term, fivesix. So this is indeed my common difference. It is two sixths. And I can go ahead and write that in lowest terms as one third. So now that I have both my first term and my common difference, I can use them to write my general formula. Remember that for an arithmetic sequence, a sub n is equal to a one plus n minus one times d. So plugging in those values here, a sub one is one half. So I have one half plus n minus one times one third. And that is my general formula, a sub n. Now this is a correct answer, but depending on your professor, you may need to simplify this further by distributing your one third or your common difference and combining any like terms. So I'm gonna go ahead and show you that here as well. So distributing our one third into our n and our negative one, this is gonna give us one half plus one third n minus one third. And we can combine our like terms, that's one half and negative one third. And I'm gonna go ahead and give those a common denominator. I'm taking one half minus one third. But I can give those a common denominator of six here, writing this as three sixths minus two sixths, which gives us here one sixth. So combining those like terms, I get that my general formula a sub n is equal to one third n plus one sixth. And this is also an acceptable answer, just further simplified. Feel free to let me know if you have any questions, and I'll see you in the next one.

Write a general formula for the arithmetic sequence given. (B) 1 2 , 5 6 , 7 6 , 3 2 , … \frac12,\frac56,\frac76,\frac32,\ldots

a n = 1 3 n + 1 6 a_{n}=\frac13n+\frac16 a n = 3 1 n + 6 1

a n = 1 3 n + 5 6 a_{n}=\frac13n+\frac56 a n = 3 1 n + 6 5

a n = − 1 3 n + 5 6 a_{n}=-\frac13n+\frac56 a n = − 3 1 n + 6 5

a n = 1 3 n + 2 6 a_{n}=\frac13n+\frac26 a n = 3 1 n + 6 2

13. Sequences, Series, and the Binomial Theorem

Arithmetic Sequences

Intermediate Algebra

13. Sequences, Series, and the Binomial Theorem

Arithmetic Sequences

Struggling with Intermediate Algebra?

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Multiple Choice

Write a general formula for the arithmetic sequence given.
(B) $\frac{1}{2}, \frac{5}{6}, \frac{7}{6}, \frac{3}{2}, \dots$

$a_{n}=\frac13n+\frac56$

$a_{n}=-\frac13n+\frac56$

$a_{n}=\frac13n+\frac26$

$a_{n}=\frac13n+\frac16$

Verified step by step guidance

Identify the general form of an arithmetic sequence, which is given by the formula: $a_n = a_1 + (n - 1)d$, where $a_n$ is the nth term, $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Determine the first term $a_1$ of the sequence from the given information or problem statement.

Find the common difference $d$ by subtracting the first term from the second term, i.e., $d = a_2 - a_1$.

Use the formula $a_n = a_1 + (n - 1)d$ to express the nth term of the sequence, substituting the known values of $a_1$ and $d$.

If asked to find a specific term or sum, plug in the value of $n$ into the formula or use the sum formula for arithmetic sequences, $S_n = \frac{n}{2} (2a_1 + (n - 1)d)$, to proceed.

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