Problem 48 - College Algebra

Question

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$1+2+3+\dots+40$$

Step-by-Step Solution

Answer

The sum \(1 + 2 + 3 + ... + 40\) can be expressed using summation notation as \(\sum_{i=1}^{40} i\).

Step 1: Identify the components of the arithmetic sequence

Firstly, identify the first term (a), the common difference (d), and the number of terms (n). Here, a = 1, d = 1 and n = 40 because the sequence starts from 1 and ends at 40, with each term having a common difference of 1.

Step 2: Write the general term of the arithmetic sequence

The general term of an arithmetic sequence can be written as \(a + (i - 1)d\) where 'i' is the index of summation. Here, substituting the identified values, the general term of the sequence is \(1 + (i - 1)1 = i\). This means any term at the 'i'th position in this sequence can be represented as 'i'.

Step 3: Write the sum using summation notation

With 1 as the lower limit and 'i' as the index of summation, the sum \(1 + 2 + 3 + ... + 40\) can be written using summation notation as: \(\sum_{i=1}^{40} i\). This sum notation accurately represents summing 'i' (where 'i' is the 'i'th term of the sequence) from 'i'=1 to 'i'=40.