Calculating the Sum of an Arithmetic Sequence: A Comprehensive Guide
A common problem in mathematics and particularly in the realm of calculus is the calculation of the sum of a series of numbers in an arithmetic sequence. This guide will take you through the step-by-step process of finding the sum of the first sixteen terms of a given sequence using the general term formula (A_n 3n^4).
Introduction to Arithmetic Sequences
Arithmetic sequences are characterized by a constant difference between consecutive terms. Typically, each term in the sequence is obtained by adding a fixed, constant value (known as the common difference) to the previous term. The general term formula for an arithmetic sequence is:
[ A_n a (n-1)d ]
However, this problem involves a more complex general term formula: ( A_n 3n^4 ). Let's explore this step by step.
Step 1: Finding the First Term
Substituting ( n 1 ) into the general term formula:
[ A_1 3 cdot 1^4 3 cdot 1 3 ]
The first term ( A_1 ) is 3.
Step 2: Finding the Sixteenth Term
Substituting ( n 16 ) into the general term formula:
[ A_{16} 3 cdot 16^4 3 cdot 65536 196608 ]
The sixteenth term ( A_{16} ) is 196608.
Step 3: Calculating the Sum of the First Sixteen Terms
The sum of the first ( n ) terms of an arithmetic sequence can be found using the formula:
[ S_n frac{n}{2} (A_1 A_n) ]
Substituting ( n 16 ), ( A_1 3 ), and ( A_{16} 196608 ).
[ S_{16} frac{16}{2} (3 196608) 8 cdot 196611 1572888 ]
Thus, the sum of the first sixteen terms is 1572888.
Alternative Approach Using the Formula for an Arithmetic Sequence
Another method involves a different general term formula. Let's consider the sequence given by ( A_n 3n^5 ).
Step 1: Finding the First Term
[ A_1 3 cdot 1^5 3 cdot 1 3 ]
The first term ( A_1 ) is 3.
Step 2: Finding the Twelfth Term
[ A_{12} 3 cdot 12^5 3 cdot 248832 746496 ]
The twelfth term ( A_{12} ) is 746496.
Step 3: Calculating the Sum of the First Twelve Terms
Using the sum formula for an arithmetic sequence:
[ S_{12} 12 cdot frac{A_1 A_{12}}{2} 12 cdot frac{3 746496}{2} 12 cdot 373249.5 12 cdot 373249.5 4478994 ]
Thus, the sum of the first twelve terms is 4478994.
Conclusion
Both approaches demonstrate how to find the sum of terms in arithmetic sequences, showcasing the versatility of the general term formula in solving various mathematical problems. Whether you are dealing with simpler linear sequences or more complex polynomial sequences, understanding these steps will equip you with the skills to tackle a wide range of problems.