Brennan has been playing a game where he can create towns and help his empire expand. Each town he has allows him to create 1.15 times as many villagers as he had in the one before. The game gave Brennan 5 villagers to start with. Explain to Brennan how to create an equation to predict the number of villagers in any specific town. Then show how to use your equation to solve for the number of villagers he can create to live in the 15th town.

Question

D14610 · Answer

Brennan's game follows a geometric sequence, with each term being a constant multiple of the preceding one. A geometric sequence has the general form a = a1 * rn-1, where an is the nth term, a1 is the first term, r is the common ratio, and n is the number of terms.

In Brennan's scenario, a1 is 5 (the number of villagers in the first town), r is 1.15 (each town has 1.15 times the number of villagers as the one before it), and n is the town number.

So, the equation for predicting the number of villagers in any given town is:

Villagersn = 5 x 1.15n-1

To find the number of villagers in the fifteenth town, substitute n = 15 into the equation:

Villagers 15 = 5 * 1.1515-1 = 5 * 1.1514.

Calculate 1.1514 and multiply by 5 to get the number of villagers in the 15th town.