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Find the first term of the arithmetic sequence where [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 need to use the formula for the nth term of an arithmetic sequence:

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

where:
- [tex]\( a_n \)[/tex] is the nth term of the sequence.
- [tex]\( a_1 \)[/tex] is the first term.
- [tex]\( n \)[/tex] is the term number.
- [tex]\( d \)[/tex] is the common difference.

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

We want to find [tex]\( a_1 \)[/tex], the first term of the sequence.

Rearrange the formula to solve for [tex]\( a_1 \)[/tex]:

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

Substitute the given values into the equation:

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

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

Calculate the value inside the parentheses:

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

Now substitute back into the equation:

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

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

Add the numbers:

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

So, the first term of the sequence is 503.

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