Problem 51 - College Algebra

Question

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$4+\frac{4^{2}}{2}+\frac{4^{3}}{3}+\dots+\frac{4^{n}}{n}$$

Step-by-Step Solution

Answer

The given sum series in the summation notation can be expressed as \( \sum_{i=1}^{n} \frac{4^{i}}{i} \).

Step 1: Identify the pattern

Observe that in each term of the series, the base number 4 is raised to the power of the term number and divided by that same term number. The base is consistent and the exponent and divisor change with each term. The sequence starts with \( \frac{4^{1}}{1} \), continues with \( \frac{4^{2}}{2} \), \( \frac{4^{3}}{3} \), and so on upto \( \frac{4^{n}}{n} \).

Step 2: Expressing in Summation Notation

As the pattern shows, every term of the series has the consistent base, exponent and divisor governed by the term number, and continues upto some term 'n'. As per the given condition, the index of summation 'i' starts from 1. Thus, the series is represented in summation notation as \( \sum_{i=1}^{n} \frac{4^{i}}{i} \).