What is the nth term of a geometric sequence? Let A(n) be the value of the nth term of the geometric sequence.
Let n be the term number in the sequence.
Let A(1) = a be the first term of the sequence.
Then, the function rule for the nth term is A(n) = a × r^{n-1}
Notice that when n = 1,
A(1) = a × r^{1-1}
A(1) = a × r^{0}
A(1) = a × 1 = a
Consider the sequence 2, 10, 50, 250, 1250, ...
A(1) = 2
A(2) = 2 × 5 = 2 × 5^{1}
A(3) = 2 × 5 × 5 = 2 × 5^{2}
A(4) = 2 × 5 × 5 × 5 = 2 × 5^{3}.
.
.
A(n) = 2 × 5 × 5 × 5 × 5 ... × 5 = 2 × 5^{n-1}
Notice that each exponent is one less than its term number.
The function rule for 2, 10, 50, 250, 1250, ... is A(n) = 2 × 5^{n-1}
A woman invests some money in the stock market. Her initial investment is 4000 dollars. Suppose the amount at the beginning of the second year is 75% of the initial investment. Then, each amount is 75% of the previous amount. How much money does she have after 4 years?
Solution
A(n) = 4000 × 0.75^{n-1}
Notice that when n = 1,
A(1) = 4000 × 0.75^{1-1} = 4000 × 0.75^{0 }= 4000 × 1 = 4000
Therefore, A(2) is money after 1 year, A(3) is money after 2 years, and A(5) is money after 4 years.
A(5) = 4000 × 0.75^{5-1}
A(5) = 4000 × 0.75^{4}
A(5) = 4000 × 0.31640625
A(5) = 1265.625
After 4 years, she has 1265.625 dollars.
Jan 18, 22 08:00 AM
What is an abacus? Learn quickly and easily to use an abacus to do math.
Jan 17, 22 09:15 AM
What is the width of an object? Definition, explanation, and easy to understand real life examples.