A. Find the sum of each of the following:
1. integers from 1 to 50
2. odd integers from 1 to 100
3. even integers between 1 and 101
4. first 25 terms of the arithmetic sequence [tex]$4,9,14,19,24, \ldots$[/tex]
5. multiples of 3 from 15 to 45
6. numbers between 1 and 81 which are divisible by 4
7. first 20 terms of the arithmetic sequence [tex]$-16,-20,-24, \ldots$[/tex]
8. first 10 terms of the arithmetic sequence [tex]$10.2,12.7,15.2,17.7, \ldots$[/tex]
9. [tex]$1+5+9+\ldots+49+53$[/tex]
10. [tex]$\frac{1}{2}+\frac{3}{2}+\frac{5}{2}+\ldots+\frac{17}{2}+\frac{19}{2}$[/tex]
Answer (2)
I calculated the sums of various sequences using arithmetic formulas. Each step outlined the method used and provided the final result for each sum. This approach helps understand how to handle different types of sequences effectively.
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The problem involves finding the sum of 10 different arithmetic sequences. The formula S n = 2 n ( a 1 + a n ) is used, where S n is the sum of the first n terms, a 1 is the first term, and a n is the n th term. In some cases, the formula a n = a 1 + ( n − 1 ) d is used to find the number of terms n or the last term a n , where d is the common difference. Each sequence is tackled individually using these formulas to find the sum.
Explanation
Problem Analysis and Strategy We are asked to find the sum of 10 different sequences. Each of these sequences is an arithmetic sequence, which means that there is a constant difference between consecutive terms. To find the sum of an arithmetic sequence, we can use the formula: S n = 2 n ( a 1 + a n ) , where S n is the sum of the first n terms, a 1 is the first term, and a n is the n th term. In some cases, we will need to find the number of terms n or the last term a n using the formula a n = a 1 + ( n − 1 ) d , where d is the common difference. Let's tackle each sequence one by one!
Examples
Understanding arithmetic sequences and their sums is crucial in many real-world scenarios. For instance, calculating the total earnings from a job with regular raises, predicting the total distance covered in equal increments, or determining the total cost of items with linearly increasing prices. These problems illustrate how arithmetic sequences help in financial planning, motion analysis, and inventory management, providing a structured approach to solving practical problems involving consistent incremental changes.
