Let n>=1.
Prove by induction for all
n>=1 1+5+9+...+[4(n-1)+1]+[4n+1] = (n+1)(2n+1).

Proved

Step-by-step explanation:

Given

Required

Prove by induction

Increment n by 1 on both sides

Simplify the right hand side

Group the left hand side

Recall that

----[Given]

So; Substitute for on the left hand side

Open All Brackets

Collect Like Terms

Notice that the expression on both sides are equal;

Hence, the given expression has been proven

We know that formula for volume of cone is 1/3 πr² h where r is radius and h is height.

Now we know that r is 2 and volume is 29 so we can set up equation like this:

29 = 1/3 π (2)² h

Now we need to solve for h:

29 = 4/3 π h

Divide both sides by 4/3 π:

29 / (4/3 π) = h

87 / (4π) = h

Use calculator here

h ≈ 6.92675159

Round to nearest hundredth:

h ≈ 6.93

Final 6.93