To solve the series [tex]\(\sum_{n=1}^{10} (2n + 1)\)[/tex], follow these detailed, step-by-step instructions:
1. Identify the series pattern:
The given series is [tex]\(\sum_{n=1}^{10} (2n + 1)\)[/tex]. This means you will sum the terms of the form [tex]\(2n + 1\)[/tex] from [tex]\(n = 1\)[/tex] to [tex]\(n = 10\)[/tex].
2. Write down the terms:
Let’s enumerate the terms for each [tex]\(n\)[/tex] value from 1 to 10:
- For [tex]\(n = 1\)[/tex]: [tex]\(2 \times 1 + 1 = 3\)[/tex]
- For [tex]\(n = 2\)[/tex]: [tex]\(2 \times 2 + 1 = 5\)[/tex]
- For [tex]\(n = 3\)[/tex]: [tex]\(2 \times 3 + 1 = 7\)[/tex]
- For [tex]\(n = 4\)[/tex]: [tex]\(2 \times 4 + 1 = 9\)[/tex]
- For [tex]\(n = 5\)[/tex]: [tex]\(2 \times 5 + 1 = 11\)[/tex]
- For [tex]\(n = 6\)[/tex]: [tex]\(2 \times 6 + 1 = 13\)[/tex]
- For [tex]\(n = 7\)[/tex]: [tex]\(2 \times 7 + 1 = 15\)[/tex]
- For [tex]\(n = 8\)[/tex]: [tex]\(2 \times 8 + 1 = 17\)[/tex]
- For [tex]\(n = 9\)[/tex]: [tex]\(2 \times 9 + 1 = 19\)[/tex]
- For [tex]\(n = 10\)[/tex]: [tex]\(2 \times 10 + 1 = 21\)[/tex]
3. List the terms of the series:
The series becomes [tex]\(3, 5, 7, 9, 11, 13, 15, 17, 19, 21\)[/tex].
4. Calculate the sum of the series:
- [tex]\(3 + 5 = 8\)[/tex]
- [tex]\(8 + 7 = 15\)[/tex]
- [tex]\(15 + 9 = 24\)[/tex]
- [tex]\(24 + 11 = 35\)[/tex]
- [tex]\(35 + 13 = 48\)[/tex]
- [tex]\(48 + 15 = 63\)[/tex]
- [tex]\(63 + 17 = 80\)[/tex]
- [tex]\(80 + 19 = 99\)[/tex]
- [tex]\(99 + 21 = 120\)[/tex]
The cumulative sums at each step are as follows: [tex]\(3, 8, 15, 24, 35, 48, 63, 80, 99, 120\)[/tex].
Therefore, the sum of the series [tex]\(\sum_{n=1}^{10} (2n + 1)\)[/tex] is [tex]\(120\)[/tex].
