When the first n positive integers are added together, their sum is given by
1/2n(n+1)
Find the value of n for which the sum is 105..
Can you show me a step by step please?
set 1/2n(n+1) = 105
multiply by 2
n(n+1) = 210
n^2 + n - 210 = 0
(n+15)(n-14) = 0
n = -15 or 14, but n has to be a positive integer, so
there are 14 terms
Thankyou
Sure! Let's solve the equation step by step:
Step 1: Set up the equation:
We are given the equation 1/2n(n+1) = 105. We want to find the value of n that makes this equation true.
Step 2: Multiply both sides of the equation by 2:
2 * (1/2n(n+1)) = 2 * 105
This simplifies to n(n+1) = 210.
Step 3: Rewrite the equation in standard form:
n^2 + n = 210
Step 4: Rearrange the equation:
n^2 + n - 210 = 0
Step 5: Factorize the equation:
(n + 15)(n - 14) = 0
Step 6: Apply zero product property:
n + 15 = 0 or n - 14 = 0
Step 7: Solve for n:
n = -15 or n = 14
Step 8: Check the solutions:
Since we are looking for positive integers, n cannot be -15. Therefore, the only valid solution is n = 14.
So, the value of n for which the sum is 105 is n = 14.
Sure! To find the value of n for which the sum of the first n positive integers is 105, we can set up the equation:
1/2n(n + 1) = 105
To solve this equation, we can follow these steps:
Step 1: Multiply both sides of the equation by 2 to eliminate the denominator:
2 * (1/2n(n + 1)) = 2 * 105
Simplifying, we have:
n(n + 1) = 210
Step 2: Expand the equation:
n^2 + n = 210
Step 3: Rearrange the equation to bring all terms to one side:
n^2 + n - 210 = 0
Step 4: Factorize the quadratic equation:
(n + 15)(n - 14) = 0
Setting each factor equal to zero, we get:
n + 15 = 0 or n - 14 = 0
Solving each equation individually:
For n + 15 = 0, subtracting 15 from both sides, we get n = -15.
For n - 14 = 0, adding 14 to both sides, we get n = 14.
Step 5: Check if the solutions are valid by substituting them back into the original equation.
When n = -15, the sum of the first n positive integers does not make sense as we cannot have a negative value for the number of positive integers. Therefore, -15 is not a valid solution.
When n = 14, substituting into the original equation, we have:
1/2 * (14)(14 + 1) = 105
1/2 * 14 * 15 = 105
7 * 15 = 105
105 = 105
Therefore, n = 14 is the valid solution where the sum of the first 14 positive integers equals 105.