The formula for the \( n \)-th term of a geometric sequence is:
["The Formula for the ( n )-th Term of a Geometric Sequence: Understanding Geometric Progressions", "A geometric sequence is a type of mathematical progression where each term after the first is found by multiplying the previous term by a constant, called the common ratio. Geometric sequences appear in diverse fields such as finance, biology, physics, and computer science, making understanding their formula essential for students, educators, and professionals alike.", "### What Is a Geometric Sequence?", "A geometric sequence begins with an initial term ( a ), followed by terms calculated by repeatedly multiplying by a common ratio ( r ). For example, if ( a = 3 ) and ( r = 2 ), the sequence starts:\n3, 6, 12, 24, 48, …\nEach term is ( r ) times larger than the one before.", "### The Formula for the ( n )-th Term", "The formula for the ( n )-th term of a geometric sequence is:", "[\na_n = a \cdot r^{n-1}\n]", "Where:\n- ( a_n ) = the ( n )-th term of the sequence\n- ( a ) = the first term (also called the initial term)\n- ( r ) = the common ratio between successive terms\n- ( n ) = the term’s position in the sequence, a positive integer (( n = 1, 2, 3, \dots ))", "### How This Formula Works", "The general form ( a_n = a \cdot r^{n-1} ) reflects how multiplication builds up through powers of ( r ):", "- At ( n = 1 ): ( a_1 = a \cdot r^{0} = a ) (the first term)\n- At ( n = 2 ): ( a_2 = a \cdot r^{1} = ar ) (second term)\n- At ( n = 3 ): ( a_3 = a \cdot r^{2} = ar^2 ) (third term), and so on.", "### Example Applications", "- Population Growth: If a bacterial colony doubles every hour and begins with 100 cells, the population after ( n ) hours is:\n ( a = 100, , r = 2 \Rightarrow a_n = 100 \cdot 2^{n-1} )\n After 3 hours: ( 100 \cdot 2^{2} = 400 ) cells.", "- Compound Interest: In finance, money invested at compound interest follows a geometric sequence. For principal ( P ), rate ( R ) per period, and ( n ) periods:\n ( A_n = P(1 + R)^{n-1} ), matching our formula.\n For ( P = $1000, R = 0.05 ) (5%), after 2 years:\n ( 1000 \cdot 1.05^{1} = $1050 ), next: ( 1000 \cdot 1.05^{2} = $1102.50 ).", "### Why Mastering This Formula Matters", "- Predict Trends: Use it to forecast values in exponential growth/decay scenarios.\n- Solve Problems Efficiently: Avoid tedious multiplication by applying the formula directly.\n- Build Advanced Skills: Forms the foundation for sequences, series, and concepts in calculus and financial mathematics.", "### Conclusion", "The formula ( a_n = a \cdot r^{n-1} ) elegantly captures the behavior of geometric sequences, unlocking powerful insights across science, engineering, economics, and everyday life. Whether calculating compound interest or modeling natural phenomena, understanding this formula empowers precise and insightful mathematical reasoning.", "---", "Keywords: geometric sequence formula, ( n )-th term geometric sequence, common ratio, mathematical formula, exponential growth, geometric progression, algebra formula, sequence theory, finance formula, compound interest, biometry model."]
