Answer:
In Exercises 1-15 use mathematical induction to establish the formula for n ≥ 1.
1. 1
2 + 22 + 32 + · · · + n
2 =
n(n + 1)(2n + 1)
6
Proof:
For n = 1, the statement reduces to 12 =
1 · 2 · 3
6
and is obviously true.
Assuming the statement is true for n = k:
1
2 + 22 + 32 + · · · + k
2 =
k(k + 1)(2k + 1)
6
, (1)
we will prove that the statement must be true for n = k + 1:
1
2 + 22 + 32 + · · · + (k + 1)2 =
(k + 1)(k + 2)(2k + 3)
6
. (2)
The left-hand side of (2) can be written as
1
2 + 22 + 32 + · · · + k
2 + (k + 1)2
.
In view of (1), this simplifies to:
Answer:
7/2 and 3
Step-by-step explanation:
using quadratic formula
{-(-13) +- √(-13)^2 - 4(2)(21) } /2(2)
-(-13)+1 / 4 = 7/2
-(-13)-1 / 4 = 3
Answer:
39 = r-13
Step-by-step explanation:
Answer: 6cm
Step-by-step explanation:
Area of a triangle is 1/2bh
B= base
H= height
Area = 30cm
Substitute the values into the formula
Area= 1/2bh
30=1/2×10×h
30=5h
H=30/5
H= 6cm
Height of the triangle is 6cm
