Question

a grocer stacks oranges in a square pyramid. each orange sits on the 4 oranges below it. so, the top layer has 1 orange and the layer below it has 4 oranges. the layer below that has 9 oranges. the total number of oranges required for 1 layer is 1. the total number of oranges required for 2 layers is 5. the total number of oranges required for 3 layers is 14. Write a recursive formula for the sequence. how many oranges are required for 10 layers?

Answer 1

The recursive formula for the sequence is a_n = a_n-1 + n^2. Using this formula, the total number of oranges required for 10 layers is 385 oranges.

Step-by-step explanation:

To find a recursive formula for the sequence, we can observe that each layer requires an additional square of oranges compared to the previous layer. So, the number of oranges required for a given layer can be calculated by adding the number of oranges in the previous layer plus the square of the layer number.

Therefore, the recursive formula is given by:

an = an-1 + n^2

Using this formula, we can calculate the number of oranges required for 10 layers:

a10 = a9 + 10^2

a10 = a8 + 9^2 + 10^2

a10 = a7 + 8^2 + 9^2 + 10^2

...

a10 = a1 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2

Calculating this sum, we find that the total number of oranges required for 10 layers is 385 oranges.

Answer 2

The top layer has 1 orange and the layer below has 4 oranges, the layer below has 9 oranges, the total number of oranges for each layer:

Layer 1 = 1 orange
Layer 2 = 4 oranges
Layer 3 = 9 oranges
Total: 14 oranges.

The number of oranges required for 10 layers can be solved using an arithmetic sequence.