SOLVED EXAMPLES 28 TO 32

Ex. 28 a number when successively divided by 1,5 and 8 leaves remainders 1, 4 and 7 respectively. Find the respective remainders if the order of divisors be reversed.

Sol.

Z=(8x1+7) =15,y =(5xz+4)=(5z15 +4) =79 X=(3y + 1) = (3x79 +1) = (237+1) = 238

                              3     x

                              5    y-1

                              8   z-4

                              8    238

                               5   29-6

                               3    5-4

                                       1-2

Hence, the respective remainders are 6, 4, 2.

Results on some series (formulae)

(I)                 (1 + 2+ 3+ …..+) = ½ n (n+1)

       (II)           (1²+2²+3²+….+n²)= 1/6 (n+1) (2n +1)

(II)               (1³+2³+3³+…..+n³)   =1/4n³(n+1)²

(iv)              Arithmetic progression (A.P)

 A, a + d, a +2d, a + 3d, Are said to be in A.P.  in which first term = a and common difference =d.

Let the n the term be than lt last term=tn=l. then

I.                    Nth term = a + (n-1) d

II.                  Sum of n terms =n/2 [2a + (n – 1) d]

III.                Sum of n terms = n/2 (a+L), where I is the last term.

(I)                 Geometric progression (G.P)

 A, ar, ar², ar³,… are said to be in G.P. in which first term = a and common ratio = r

I.                    Nth term = arᶯ-1

II.                  Sum of n terms =     a (1-rᶯ),                     when r <1

                                                        (1-r)

                                                    a (rᶯ - 1)

                                                       (r-1)

                                                                                             , when r <1

Ex . 29. How many natural numbers between 17 and 80 are divisible by 6?

 Sol. These numbers are 18, 24, 36,…., 78.

This is an A.P in which a =18, d =(24-18) =6and l =78.

Let the number of these terms be n. Then,

tn =78  a+(n-1) d =78

18+ (n-1) x 6= 78 (n-1) x6=60 (n -1) =10 n=11

Required number of numbers –=11.

Ex. 30. Find the sum of all even natural numbers less than 75.

Sol required sum =2+4+6+…+74

This is an A.P. in which a =2,d= 2, d=(4-2) =2,1 =74

Clearly, n =37.

Required sum = n/2 ( a+l) = 37/2x (2+74)=(37x38)

                            =37x(40 – 2) (37 x 40) –(37x2)

=(1480 – 4  = 1406.

Ex. 31. (6 + 15+ 24 +33 +  …+ 105)  =?

Sol. Given series in an A.P. in which a =6, d=(15-6) = 9and l = 105

Let the number of terms is it be n. Then,

A+(n-1) d =105 6 ++(n-1)x9=105

(n-1) x9=99 (n-1=11n=12

Required sum = n/2 (a+l)= 12/2 x(6+105) = (6x111) =666.

Ex. 32. Find the sum (2+ 2²+ 2²+ 2⁴+ … + 2¹⁰)

Sol this is a G.P. in which a =2, r=2²/2=4/2=2

Required sum = a(rᶯ -1) =   2x (2¹⁰ -1)

                             (r-1)                 (2-1)                         = (2x1023)= 2046.

A= 2046