Question

PLEASE HELP AND SHOW ALL WORK

Answer 1

Answer:

1. The given series is

  4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = $\frac{[4(4n+1)(8n+7)]}{6}$

For n=1

L.H.S=4.6=24

R.H.S=[4×5×15]÷6

       =300÷6

       =50

So, for n=1,

L.H.S≠ R.H.S

Since the given expression is true for n=1 ,

So , the given series is untrue.

we should replace R.H.S by=4(n+1)(n+2)(4n-3)²

2.

12+42+72+.......+(3 n -2)2=$\frac{n(6 n²-3 n- 1)}{2}$

For n=1,

L.H.S=12

R.H.S=1×(6-3-1)/2

        =2/2

       =1

As L.H.S≠ R.H.S

We should Replace R.H.S by [(3 n-1)(3 n-2)]2

3.The given sequence is

2+4+6+....+2n=n(n+1)

L.H.S

P(1)=2

R.H.S

1×(1+1)

=1×2

=2

( b)  L.H.S

P(n)=2+4+6+.....2 k

This is an A.P having n terms.

$S_{n}=[tex]\frac{n}{2}\times\text{[first term + last term]}$

tex]S_{n}=$\frac{n}{2}\text [{2+2n}]$

               = n(n+ 1)

R.H.S=n(n+1)

So, P(k)=k(k+1)

(c) P(k+1)=2+4+6+.......+2(k+1)

This is an A.P having (k+1) terms.

$S_(k+1)=\frac{k+1}{2}[2+2k+2]$

               =(k+1)(k+2)

So, P(k+1)= (k+1)(k+2)

Answer 2

1]
4*6+5*7+6*8+.....+4n(4n+2)=4(4n+1)(8n+7)/6
If we choose n=1, then 4*6=24 but 4(4*1+1)(8*1+7)/6=50. This implies that the general trm for the pattern shown is wrong. It should have been (n+3)(n+5) and not 4n(4n+2).

2] 12+42+72+.......+(3n-2)2=n(6n²-3n-1)/2
Let's set n=1, this means that 12=12.
But n(6n²-3n-1)/2
=1(6*1²-3*1-1)/2
=(6-3-1)/2
=2/2
=1
This shows that the general term is incorrect. It should have been (30n-18) which when simplified we get 6(5n-3). Even if we get to correct the left hand side the sequence will still not be equal to what's on the right given n=1.

3] 2+4+6+....+2n=n(n+1)
P(1):2=1(1+1)
P(m):2+4+6+..+2m=m(m+1)
P(m+1):2(k+1)=(k+1)(k+2)