1.1 Given the Arithmetic sequence: \( 10+15+20 \) 1.1 .1 How many terms are there in the series? 1.12 . Write down the series in sigma notation 1.13 Calculate the number of the series.
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In the given arithmetic sequence, the first term is 10 and the common difference is 5 (15 - 10 = 5). To find the number of terms, we can express the nth term using the formula \( a_n = a_1 + (n-1)d \). Setting \( a_n = 20 \), we solve for n: \( 20 = 10 + (n-1)5 \) which gives \( n = 3 \). So, there are 3 terms in the series. Using sigma notation, the series can be represented as \( \sum_{n=1}^{3} (10 + (n-1)5) \). This sum effectively captures the sequence's pattern and can be useful when working with larger series, allowing you to express complex summations in a compact form. For calculating the sum of the series, you can also use the formula for the sum of an arithmetic series: \( S_n = \frac{n}{2} (a_1 + a_n) \). Substituting the values gives \( S_3 = \frac{3}{2} (10 + 20) = 45 \). Thus, the sum of the series is 45, making it a perfect example of how arithmetic sequences work!
