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Here is the solution to (a) in English:
1. (a) An arithmetic sequence is composed of numbers exactly divisible by 4, with the last number being 928. The first number is 4.
Step 1: Determine the common difference. Since all numbers in the sequence are exactly divisible by 4, the common difference between consecutive numbers must be 4. The common difference is .
Step 2: Determine the number of terms in the series. We use the formula for the -th term in an arithmetic sequence: . We have , , and . \begin{align*} 928 &= 4 + (n-1)4 \\ 928 - 4 &= (n-1)4 \\ 924 &= 4(n-1) \\ \frac{924}{4} &= n-1 \\ 231 &= n-1 \\ n &= 231 + 1 \\ n &= 232 \end{align*} The number of terms in the series is .
Step 3: Determine the sum of all terms. We use the formula for the sum of an arithmetic series: . We have , , and . \begin{align*} S_{232} &= \frac{232}{2}(4 + 928) \\ S_{232} &= 116(932) \\ S_{232} &= 108112 \end{align*} The sum of all terms is .
Step 4: Determine the 80th term. We use the formula for the -th term: . We want to find the 80th term, so . We have and . \begin{align*} a_{80} &= 4 + (80-1)4 \\ a_{80} &= 4 + (79)4 \\ a_{80} &= 4 + 316 \\ a_{80} &= 320 \end{align*} The 80th term is .
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1. (a) An arithmetic sequence is composed of numbers exactly divisible by 4, with the last number being 928.
This languages problem is solved step by step below, with detailed explanations to help you understand the method and arrive at the correct answer.