A person borrow $8000 at 2.76% simple interest per annum.The principal and interest are to be paid in 10 monthly installments.If each installmeny is double the preceding one, the value of first and last installments are??A person borrow $8000 at 2.76% simple interest per annum.The principal and interest are to be paid in 10 monthly installments.If each installmeny is double the preceding one, the value of first and last installments are??
Principal and interest after 8 months
=8000(1+0.0276/12*10)=8184
Let a=initial repayment amount (n=1), then the repayment amount at month n
a(n)=a*r^(n-1) where r=2
Total repayment amount
Sum=a(1)+a(2)+...+a(10)
=a(r^10-1)/(r-1)
=1023a
Equate total amount owing with amount paid:
8184=1023a
Solve for a.
Hah! Looks like this person is getting themselves into a bit of a pickle with all those numbers and installments. But don't worry, I'm here to help!
To find the value of the first and last installments, let's break it down step by step.
First, let's figure out the total amount of interest accrued over the course of 10 months. We'll use the simple interest formula:
Interest = Principal × Rate × Time
Interest = $8000 × 0.0276 × (10/12)
Calculating that, we get an interest of $183.
Next, let's calculate the total amount to be paid back (principal + interest):
Total amount = Principal + Interest
Total amount = $8000 + $183
Total amount = $8183
Now, let's set up the equation for the installments.
Let the first installment be x. We know that each subsequent installment is double the preceding one, so the second installment would be 2x, the third installment would be 4x, and so on.
The sum of the installments can be calculated using the formula for the sum of an arithmetic series. In this case, we have 10 installments, so the formula becomes:
Sum = (n/2)(first term + last term)
8183 = (10/2)(x + (2^10) * x)
Simplifying that gives us:
8183 = 5.5x + 2047x
Combining like terms:
8183 = 2052.5x
Now, divide both sides by 2052.5 to find the value of x:
x = 8183 / 2052.5
And after doing the math, we find:
x = 4
So, the value of the first installment is $4.
Since each installment is double the preceding one, the value of the last installment would be:
Last installment = 2 * (4 * 2^9)
Calculating that gives us:
Last installment = $8,192
So, the value of the first installment is $4 and the value of the last installment is $8,192.
I hope that answers your question! And remember, when it comes to money matters, always keep your clown nose on and calculate with a smile!
To find the value of the first and last installments, we can use the formula for the sum of an arithmetic series.
The formula for the sum of an arithmetic series is given by: S = (n/2)(a + l), where S is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
In this case, the sum of the series is $8000, which is the principal amount borrowed. The number of terms is 10, and we need to find the first and last terms of the series.
Let's use the formula to find the value of the first installment:
S = (n/2)(a + l)
8000 = (10/2)(a + l)
8000 = 5(a + l)
Next, we'll use the fact that each installment is double the preceding one. Let's denote the first installment as a, and the last installment as l. We can express the other installments in terms of a:
1st installment = a
2nd installment = 2a
3rd installment = 4a
4th installment = 8a
...
nth installment = 2^(n-1) * a
Since the last installment is double the preceding one, we have:
l = 2^(n-1) * a
Now, substitute the expression for l into our equation for the sum of the series:
8000 = 5(a + 2^(n-1) * a)
Simplify the equation:
8000 = 5a + 5 * 2^(n-1) * a
8000 = 5a(1 + 5 * 2^(n-1))
To find the value of the first installment, we need to solve for a. Rearrange the equation:
5a = 8000 / (1 + 5 * 2^(n-1))
a = 8000 / (5 * (1 + 5 * 2^(n-1)))
Now, we can substitute the value of a into the expression for the last installment:
l = 2^(n-1) * a
l = 2^(n-1) * (8000 / (5 * (1 + 5 * 2^(n-1))))
So, the value of the first installment is 8000 / (5 * (1 + 5 * 2^(n-1))), and the value of the last installment is 2^(n-1) * (8000 / (5 * (1 + 5 * 2^(n-1)))).
However, we need to know the value of n (the number of terms) to calculate the exact values of the first and last installments. Please provide the value of n, and I can help you find the answer.
To find the value of the first and last installments, we need to understand the calculation involved in computing the monthly installment amount.
1. Let's start by calculating the monthly interest rate. Since the interest is stated as an annual rate, we need to divide it by 12 to get the monthly rate.
Monthly interest rate = (annual interest rate) / (number of months in a year) = 2.76% / 12 = 0.23% per month
2. Next, we can determine the monthly installment amount using the formula for simple interest:
Monthly installment amount = (Principal + Total Interest) / Number of months
3. The total interest can be calculated by multiplying the principal by the interest rate and the number of months.
Total Interest = Principal * Interest Rate * Number of Months
As the interest is simple interest, we can calculate the total interest as:
Total Interest = Principal * (Interest Rate / 100) * Number of Months
4. Now, we can substitute the values into the formula to find the monthly installment amount:
Monthly installment amount = (Principal + Total Interest) / Number of months
5. Since each installment is double the preceding one, we can establish the following relationship:
Second installment = 2 * First installment
Third installment = 2 * Second installment
6. We can use this relationship to compute the value of the first installment and then find the value of the last installment.
Let's do the calculations:
Principal = $8000
Interest Rate = 2.76% per annum
Number of months = 10
Interest Rate per month = 0.23%
Total Interest = Principal * (Interest Rate / 100) * Number of Months
= $8000 * (2.76 / 100) * 10
= $2208
Monthly installment amount = (Principal + Total Interest) / Number of months
= ($8000 + $2208) / 10
= $1020.80 (approx.)
Now, we can find the value of the first installment using the given relationship:
First installment = $1020.80 / (2 + 1 + 1/2 + 1/4 + ... + 1/256)
= $1020.80 / (2 * (1 - (1/2)^8))
= $1020.80 / (2 * 0.99609375)
= $513.96 (approx.)
To find the value of the last installment, we need to double the previous installment:
Last installment = 2 * $513.96
= $1027.92 (approx.)
So, the value of the first installment is approximately $513.96, and the value of the last installment is approximately $1027.92.