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Find the 57th term of the sequence:

Given sequence: [tex]\(13, 8, 3, -2, -7, -12, \ldots\)[/tex]

1. Enter the common difference:

[tex]\[ \text{common difference} = 8 - 13 \][/tex]

2. Use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:

[tex]\[ a_n = a_1 + (n-1)d \][/tex]

3. Find the 57th term:

[tex]\[ a_{57} = 13 + (57-1)(-5) \][/tex]



Answer :

GinnyAnswer
To find the 57th term of the given arithmetic sequence, we start by identifying the components of the sequence:

1. The first term ([tex]\(a\)[/tex]) is [tex]\(13\)[/tex].
2. The common difference ([tex]\(d\)[/tex]) is calculated as [tex]\(3 - 8 = -5\)[/tex].

The formula to find the [tex]\(n\)[/tex]th term ([tex]\(a_n\)[/tex]) of an arithmetic sequence is:
[tex]\[ a_n = a + (n - 1) \cdot d \][/tex]

Let's plug in the values:
- [tex]\(a = 13\)[/tex] (the first term)
- [tex]\(d = -5\)[/tex] (the common difference)
- [tex]\(n = 57\)[/tex] (the position of the term we want to find)

Substitute these values into the formula:
[tex]\[ a_{57} = 13 + (57 - 1)\cdot (-5) \][/tex]

Simplify the expression inside the parentheses:
[tex]\[ a_{57} = 13 + 56 \cdot (-5) \][/tex]

Now, multiply [tex]\(56\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[ 56 \cdot (-5) = -280 \][/tex]

Add this result to the first term:
[tex]\[ a_{57} = 13 + (-280) \][/tex]

This simplifies to:
[tex]\[ a_{57} = 13 - 280 = -267 \][/tex]

So, the common difference is [tex]\(-5\)[/tex], and the 57th term of the sequence is [tex]\(-267\)[/tex].

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