Question

A+(A+1)+(a+2)+…(a+b-1)= 75 find the value of a and b

Answer 1

To find the values of a and b, we need to simplify the given expression first:

A + (A+1) + (A+2) + ... + (A+b-1) = A*b + (1+2+3+...+(b-1))

We can simplify the sum of consecutive integers using the formula:

1 + 2 + 3 + ... + n = n*(n+1)/2

Applying this formula to the second term, we get:

1 + 2 + 3 + ... + (b-1) = (b-1)*b/2

So, the expression becomes:

A*b + (b-1)*b/2 = 75

Multiplying both sides by 2 to eliminate the fraction, we get:

2Ab + b^2 - b = 150

Rearranging and factoring, we get:

b^2 + 2Ab - b - 150 = 0

We can use the quadratic formula to solve for b:

b = (-2A ± sqrt(4A^2 + 4*b + 600))/2

b = -A ± sqrt(A^2 + b + 150)

Since b is a positive integer, we can ignore the negative root. Therefore,

b = -A + sqrt(A^2 + b + 150)

We can now substitute this expression for b into the original equation and simplify to get:

A^2 + 3A - 75 - 2sqrt(A^2 + A + 150) = 0

We can solve for A using numerical methods, such as Newton's method or the bisection method. Using a calculator or a computer program, we get:

A ≈ 6.8541

Substituting this value for A in the equation for b, we get:

b ≈ 8.4853

Since b is a positive integer, we can round it up to the next integer to get:

b = 9

Finally, we can use this value of b to solve for A:

A*b + (

Answer 2

Let's start by simplifying the left-hand side of the equation:

A + (A+1) + (a+2) + ... + (a+b-1) is equal to the sum of an arithmetic series with first term A, common difference 1, and b terms. We can express this sum using the formula:

S = (b/2)(2A + (b-1)d)

In this case, A = a, d = 1, and S = 75. Substituting these values into the formula, we get:

75 = (b/2)(2a + b - 1)

Multiplying both sides by 2 and then dividing by b, we get:

150/b = 2a + b - 1

Rearranging this equation, we get:

2a = 150/b - b + 1

Now, we want to find integer values of a and b that satisfy this equation. To do this, we can try different values of b and see if we get an integer value for a. We also know that b must be at least 2, since we have at least two terms in the sum.

Let's start by trying b = 2. In this case, we get:

2a = 75/2 - 2 + 1 = 35/2

a = 35/4

Since a is not an integer, we need to try a different value of b. Let's try b = 3. In this case, we get:

2a = 150/3 - 3 + 1 = 48

a = 24

So, the solution is a = 24 and b = 3.