High School

Find the first term of the arithmetic sequence in which [tex]$a_{61} = 293$[/tex] and the common difference is [tex]-3.5[/tex].

A. 503
B. 83
C. -210
D. -83

Answer :

To find the first term of the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:

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

Here:
- [tex]\( a_n \)[/tex] is the nth term.
- [tex]\( a_1 \)[/tex] is the first term, which we need to find.
- [tex]\( n \)[/tex] is the term number.
- [tex]\( d \)[/tex] is the common difference.

Given:
- [tex]\( a_{61} = 293 \)[/tex] (61st term),
- The common difference [tex]\( d = -3.5 \)[/tex],
- [tex]\( n = 61 \)[/tex].

We need to rearrange the formula to solve for the first term [tex]\( a_1 \)[/tex]:

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

Now, plug in the given values:

[tex]\[ a_1 = 293 - (61 - 1) \times (-3.5) \][/tex]

Calculate the expression inside the parentheses:

[tex]\[ 61 - 1 = 60 \][/tex]

So the equation becomes:

[tex]\[ a_1 = 293 - 60 \times (-3.5) \][/tex]

Calculate the multiplication:

[tex]\[ 60 \times (-3.5) = -210 \][/tex]

Now, substitute back into the equation:

[tex]\[ a_1 = 293 - (-210) \][/tex]

Subtracting a negative is the same as adding:

[tex]\[ a_1 = 293 + 210 \][/tex]

Finally, add the numbers:

[tex]\[ a_1 = 503 \][/tex]

The first term of the arithmetic sequence is [tex]\( 503 \)[/tex].

Other Questions