Answer:
1. The given series is
4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = ![\frac{[4(4n+1)(8n+7)]}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B%5B4%284n%2B1%29%288n%2B7%29%5D%7D%7B6%7D)
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=
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]}](https://tex.z-dn.net/?f=S_%7Bn%7D%3D%5Btex%5D%5Cfrac%7Bn%7D%7B2%7D%5Ctimes%5Ctext%7B%5Bfirst%20term%20%2B%20last%20term%5D%7D)
tex]S_{n}=![\frac{n}{2}\text [{2+2n}]](https://tex.z-dn.net/?f=%5Cfrac%7Bn%7D%7B2%7D%5Ctext%20%5B%7B2%2B2n%7D%5D)
= 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]](https://tex.z-dn.net/?f=S_%28k%2B1%29%3D%5Cfrac%7Bk%2B1%7D%7B2%7D%5B2%2B2k%2B2%5D)
=(k+1)(k+2)
So, P(k+1)= (k+1)(k+2)
Answer:
We cant see the graph stup!d
Step-by-step explanation:
Answer:
1,58,18,400
Step-by-step explanation:
1st digit can have the values 1-9 (9 distinct values)
2nd digit can have the values 0-9 (10 distinct values)
3rd digit can have the values 0-9 (10 distinct values)
1st letter can have the value A-Z (26 distinct values)
2nd letter can have the value A-Z (26 distinct values)
3rd letter can have the value A-Z (26 distinct values)
Total number of different plates possible = 9*10*10*26*26*26
=1,58,18,400