the formula 1/2(n^2+n) finds the sum of the first n positive integers. find the sum of the first 5 integers, then use the formula to prove you got the same answer. finally use the formula to find the sum f the first 14 integers. answer asap for brainliest worth 30 points.

Induction is done by demonstrating that if the condition is true for some n then it must also be true for n+1. If you then show that the condition is true for n=0 then it must be true for all n>0. For this problem:

Step 1: n=1

The sum of the first 1 odd numbers is 1. 12=1. Therefore the condition holds for n=1.

Step 2: induction

If the sum of the first n odd numbers is n2 then the sum of the first n+1 integers is n2+(2n+1)=(n+1)(n+1)=(n+1)2

So the condition is also true for n+1.

Step 3: conclusion

Since the we have shown that the condition is true for n=1 and we have shown that if it is true for n then it is also true for n+1 then it follows by induction that it is true for all n≥1.

Step-by-step explanation:

sorry if wrong

infinitely many solutions

step-by-step explanation:

we have been given the equation;

2(2x + 3) = -4 + 2(2x + 5)

we need to determine the value of x.

2(2x + 3), we expand this expression to obtain 4x + 6

similarly 2(2x + 5) will become 4x + 10

now we have;

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 10 - 4

4x + 6 = 4x + 6

the expression on the left hand side is identical to the one on the right hand side. therefore, we are looking at just a single straight line whose equation is;

y = 4x + 6

consequently, we shall have infinitely many solutions