Problem 87 - Precalculus with Limits

Question

In Exercises 87 - 92, use summation notation to write the sum. \( 10 + 30 + 90 + \cdots + 7290 \)

Step-by-Step Solution

Answer

The sum \( 10 + 30 + 90 + \cdots + 7290 \) can be expressed in summation notation as \( \sum_{i=1}^{5} 10 \cdot 3^{(i-1)} \).

Step 1: Identify the Initial Term

The first term of the sequence is 10. So, the initial term (a) of the given geometric sequence is 10.

Step 2: Find the Common ratio

The common ratio (r) of a geometric sequence can be obtained by dividing any term by its preceding term. Doing this for the given sequence, we get: \( r = \frac{30}{10} = 3\)

Step 3: Write the General Term

We know that the nth term (Tn) of a geometric sequence can be expressed as \( a \cdot r^{(n-1)} \). Replacing a with 10 and r with 3, the general term \( Tn = 10 \cdot 3^{(n-1)} \)

Step 4: Express in Summation Notation

Now that we have the general term, we can finally express this sequence in summation notation. A geometric sequence can be represented in summation notation as \( \sum_{i=1}^{k} a \cdot r^{(i-1)} \) where i is the index of summation, k is the last term index. Considering the sequence continues till 7290, which is the \( T5 = 10 \cdot 3^{(5-1)} \) term, the desired summation notation will be \( \sum_{i=1}^{5} 10 \cdot 3^{(i-1)} \)

Key Concepts

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This creates a pattern of numbers increasing (or decreasing, if the ratio is between zero and one) by the same multiple each time. In our given exercise, the sequence starts with 10 and increases by a factor of 3 each time, which forms a geometric sequence due to this consistent multiplication.

Common Ratio

The common ratio in a geometric sequence is the constant factor between consecutive terms. Understanding the common ratio is crucial as it helps establish the behavior of the sequence, whether it's expanding, contracting, or oscillating (if the common ratio is negative). For the sequence in the exercise, the common ratio is calculated by dividing any term by its previous term. For example, dividing the second term (30) by the first term (10) yields the common ratio, which is 3 in this case.

Nth Term of a Geometric Sequence

The nth term of a geometric sequence is important in determining any term's value based solely on its position in the sequence. It is given by the formula \( T_n = a \times r^{(n-1)} \), where \( T_n \) is the nth term, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term's position in the sequence. In our exercise, we use the first term 10 and the common ratio 3 to find any term in the sequence. For instance, the third term would be \( 10 \times 3^{(3-1)} = 90 \).

Index of Summation

The index of summation, typically represented by the letter 'i', 'k', or 'n', is the variable used in summation notation to indicate the position of terms within the sequence. This variable takes on successive integer values starting from the lower limit and going up to the upper limit of the summation. In the exercise's solution, the index of summation 'i' starts at 1 and increments by 1 until it reaches the last term's index. The summation notation effectively compresses the entire sequence into a compact mathematical expression, facilitating both its analysis and summation.