A brick staircase has a total of 16 steps The bottom step requires 112 bricks. Each successive step requires 4 fewer bricks than the prior one. How many bricks are required to build the staircase?
-224
Let's start by finding the number of bricks required for each step.
The bottom step requires 112 bricks.
The next step requires 4 fewer bricks, so it will require 112 - 4 = 108 bricks.
The next step will require 108 - 4 = 104 bricks.
And so on...
To find the number of bricks required for each step, we can use a decreasing arithmetic sequence.
We can see that the common difference between each step is 4 (since each successive step requires 4 fewer bricks).
So, the number of bricks required for each step can be represented as:
112, 108, 104, 100, ...
Now, let's find the number of bricks required for the entire staircase.
We know that there are 16 steps in total.
To find the total number of bricks, we can sum up the number of bricks required for each step.
So, we need to find the sum of the first 16 terms of the arithmetic sequence.
The formula to find the sum of the first n terms of an arithmetic sequence is given by:
Sn = (n/2)(2a + (n-1)d)
Where:
Sn = sum of the first n terms of the sequence
n = number of terms in the sequence
a = first term of the sequence
d = common difference between the terms
In this case, n = 16 (since there are 16 steps), a = 112 (the number of bricks required for the first step), and d = -4 (since each step requires 4 fewer bricks).
Let's substitute these values into the formula and calculate the sum:
Sn = (16/2)(2(112) + (16-1)(-4))
= 8(224 - 15*4)
= 8(224 - 60)
= 8(164)
= 1312
Therefore, a total of 1312 bricks are required to build the staircase.
To find out how many bricks are required to build the staircase, we need to sum up the number of bricks needed for each step.
In this case, we know that the bottom step requires 112 bricks and each successive step requires 4 fewer bricks than the previous one.
Let's break down the steps:
Step 1: Requires 112 bricks.
Step 2: Requires 112 - 4 = 108 bricks. (4 fewer bricks than the first step)
Step 3: Requires 108 - 4 = 104 bricks. (4 fewer bricks than the second step)
Step 4: Requires 104 - 4 = 100 bricks. (4 fewer bricks than the third step)
And so on...
We can see that each step requires 4 fewer bricks than the previous one. So, we can easily calculate the number of bricks required for each step using the formula: Number of bricks for step n = 112 - (n-1) * 4
Now, we need to sum up the number of bricks for all the steps from 1 to 16.
Total number of bricks required = (Number of bricks for step 1) + (Number of bricks for step 2) + ... + (Number of bricks for step 16)
We can calculate this using the formula for the sum of an arithmetic series: Sum = (n/2)(first term + last term), where n is the number of terms, and the first and last terms are the number of bricks for step 1 and step 16, respectively.
Using the formula, we have:
Sum = (16/2)(112 + (112 - (16-1) * 4))
Simplifying the expression inside the parentheses:
Sum = (8)(112 + (112 - 15 * 4))
Sum = (8)(112 + (112 - 60))
Sum = (8)(112 + 52)
Sum = (8)(164)
Sum = 1312
Therefore, 1312 bricks are required to build the entire staircase.
you are finding the sum of 16 terms of an arithmetic sequence where
a = 112
d = -4
n = 16
S16 = (16/2)[2(16) + 15(-4) ]
= .....