Question

How do you write the series \[7 + 10 + 13 + 16 + 19\] using summation notation?

Answer 1

Hint:
Here, we will first find the common difference of the given series and try to find the common relation between the given terms. Using this we will find the general term of the given series such that after substituting the values according to the position of the numbers in the series, we will be able to find the given numbers. Hence, we will be able to find the required summation notation.

Complete step by step solution:
The given series is \[7 + 10 + 13 + 16 + 19\].
Now, as we can observe, the difference between each term and its preceding term is 3.
Also, if we try to write the first term i.e. 7 with a summation of 3, then, we can write it as:
\[7 = 4 + 3\]
Also, since each term has a common difference of 3, it must contain a sub-term \[3i\]and initial term 4 which is more than \[3i\]such that \[i = 1,2,3,4,5\] because there are 5 terms.
Thus, we get the general summation as:
\[7 + 10 + 13 + 16 + 19 = \sum\limits_{i = 1}^{i = 5} {\left( {4 + 3i} \right)} \]
On substituting \[i = 1,2,3,4,5\], we get the terms \[7,10,13,16,19\] respectively.
Therefore, using summation notation the series \[7 + 10 + 13 + 16 + 19\] can be written as

\[\sum \limits_{i = 1}^{i = 5} {\left( {4 + 3i} \right)} \].
Hence, this is the required answer.

Note:
A summation means the act of adding or doing a cumulative sum of the given element by substituting the different values of the same variable in the same element and adding them together. Also, a sigma symbol ,\[\sum {} \], denotes a sum of multiple terms or elements. Now, the basic difference between a summation and a sigma is that, summation is the adding up of the given series of elements whereas, a sigma is just a mathematical symbol used to indicate this summation without stating anything. Hence, summation plays an important role for finding out the aggregate value of a given element from its lower limit to upper limit of summation.