Question

Solve the following questions with clear explanations...

Points: 25 ☺

Answer 1

16      a+18d=3(a+5d)
       a+18d=3a+15d
       2a=3d

Now. a+8d=19
        a=19-8d


     2(19-8d)=3d
    38=16d+3d
   38=19d
   d=2
  a=3
AP=3,5,7,9..

17.a+7d=41
    a+12d=61   

d=4    a=13  19th term=85


Sum=n/2(a+l)
      =19/2*(13+85)
     =19*49=931..

20.   3n²+5n
    =3n²+5n-[3(n-1)²+5(n-1)}
    =3n²+5n-3n²-3+6n-5n+5
  =6n+2   divide by 2
   ath term=3n+1

  d =3n+1-[3(n-1)+1
    =3n+1-3n+1-1
    d=1 answer.

21.  let three no.=a,a-d,a+d
  a+a-d+a+d=15
  3a=15
  a=5
  
  a²+(a-d)²+(a+d)²=93
  25+25+d² -10d +25+d²+10d=93
 2d²=93-75
 d²=9
 d=+ve or -ve 9

Answer 2

16. Let 'a' be the 1st term of the AP and 'd' be the common difference.
So 9th term --
a + 8d = 19 -------[i]

Now,A/Q
a + 18d = 3 [ a + 5d ]
a + 18d = 3a + 15d
- 2a + 3d = 0
2a - 3d = 0 -----[ii]

Equating [i] and [ii],
19d = 38
d = 2

2a - 3 x 2 = 0
2a = 6
a = 3

So the AP is 3 , 5 , 7, 9 .......

17.Let 'a' be the 1st term and 'd' be the common difference of the AP.

8th term = a + 7d = 41 -------[i]
a + 12d = 61 -----------[ii]

Solving [i] and [ii]
5d = 20
d = 4

a + 12 x 4 = 61
a = 61 - 48
a = 13

19th term, l  = a + 18d = 13 + 18 x 4 = 85
Sum = n/2 [ a + l ] = 19/2 [ 13 + 85] = 19/2 x 98 = 19 x49 = 931

18. first term,a = 108
d = 9
last term, l = 999

Let the number of terms be 'n'
So nth term = 999
999 = 108 + [n - 1]9
999 = 108 + 9n - 9
999 = 99 + 9n
9n = 900
n = 100

Sum = n/2 [ a + l] = 100/2 [ 108 + 999] = 50 x 1107 = 55350

19. Let 2k - 7 = a
k + 5 = b
3k + 2 = c

So , a + c = 2b
2k - 7 + 3k + 2 = 2 [ k + 5]
5k - 5 = 2k + 10
3k = 15
k = 5

Value of K = 5

20 . Sn = 3n² + 5n
S₁ = 3 x 1¹ + 5 x 1 = 3 + 5 = 8
S₂ = 3 x 2² + 5 x 2 = 12 + 10 =22

a₁ = 8
a₂ = 22 - 8 = 14
d = 14 - 8 = 6

So the common difference is 6

21. Let the first 3 terms be a - d , a  , a + d
So their sum --
a - d+  a + a + d = 15
3a  = 15
a  = 5

Now sum of the squares ---
[a - d]² + a² + [a + d]² = 93
a² + d² - 2ad  + a² + a² + d² + 2ad = 93
3a² + 2d² = 93
3 x 5² + 2d² = 93
75 + 2d² = 93
2d² = 18
d² = 9
d = 3 or d = -3

Now AP can be two types

1st AP -- 5, 5 + 3 , 5 + 6..... = 5 , 8 , 11,13......
2nd AP -- 5 , 5 - 3 , 5 - 6.... = 5 , 2 , -1 , -4......

Hope This Helps You