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Syllabus

sinx+sin2x+sin3x+...+sin nx=sin((n+1)/2)x .((sin nx)/2) / sin(x/2)

5+15+45....+5.3

^{n-1}= 5/2(3^{n-1})Prove that 1.2 + 2.3 + 3.4 +..............n(n+1) = 1/3 n (n+1)(n+2)

sinx + sin3x + ..............+ sin(2n-1)x = sin^2 nx / sinx

prove by PMI

1. two classes meet at the same hr.

2. two classes meet at different hrs. and 30 students are enrolled in both the courses

3. what value is shown here?

prove by using PMI that 4 raise to n + 15n - 1 is divisible by 9 .

1^4+2^4+3^4+..........+n^4 =n(n+1)(2n+1)+(3n^2+3n-1)/30. (whole divided by 30)

Prove that 2.7n + 3.5n - 5 is divisible by 24 for all n belongs to N. [ please explain with steps]

Prove that x

^{2n}-y^{2n}is divisible by x+y?1 sqare + 2 square + 3 square +.............+n square >n cube +3 . prove it by mathematical induction.

Prove that n(n+1)(n+5) is a multiple of 3

Prove by PMI

1) a+(a+d)+(a+2d)+ ......[a+(n-1)d] = n/2 [2a+(n-1)d], n E N.

2) n(n+1)(n+5) is divisible by 6 for all n E N.

3) 9 raised to n - 8n - 1 is a multiple of 64 for all n E N.

Prove that n(n+1)(n+2) is divisible by 6

Find all positive integers n such that 3

^{2n}+ 3n^{2}+ 7 is a perfect square.Using PMI Prove that 1^2 + 2^2 + ......... + n^2 n^3/3

p(n) =1+4+7+...+(3n-2)=1/2n(3n-1)

P.T by principle of mathematical induction

using induction, prove that 10

^{n }+ 3.4^{n+2}+ 5 is divisible by 9^{2n}+ 2^{3n-3}, 3^{n-1}is divisible by 25, for all n Ꜫ N.find the equation of parabola whose focus is (1,1) and tangent at the vertex is x + y = 1

prove that 1x2 + 2x2^2 + 3x2^3+..+nx2^n = (n-1)x2^n+1 +2

Prove that 11

^{n+2 }+ 12^{2n+1 }is divisible by 133 for all n belongs to N .2^5n>3^3n for n belongs to natural no.

Prove: 5

^{2n}-1 is divisible by 24 for all n Na

^{2n-1}-1 is divisible by a-1, prove by PMI.Using PMI, prove that

5

^{2n+2}-24n-25 is divisible by 576 for n belongs to N.Prove the following

1.3 + 3.5 + 5.7 + ..... +(2n - 1) (2n + 1 ) =n( 4n square + 6n -1)/2 plz explain briefly in a simple method

Q. If P(n) is statement "n

^{2}-n+41 is a prime number"' show that P(1),P(2) are true but P(41) is not true.using principle of mathematical induction prove that, 1

^{2}+ 2^{2}+....+ n^{2}(n^{3}/3) for all n belonging to natural numbersQ. Prove that 1.3+2.4+3.5+....+n(n+2) = 1/6n (n+1) (2n+7),

~~V~~nEN.how to split a cubic polynomial like k

^{3}+6k^{2}+9k+4??5+15+45+....+5.3

^{n-1}= 5/2(3^{n-1})bY PMI prove n(n+1)(2n+1) is divisible by 6

^{2n-1}+ 1 is divisible by 16 .Prove that 2.7

^{n}+ 3.5^{n}-5 is divisible by 24, for all n belongs to N....Plzzz dnt tell to refer textbook as frm dat also its not clear to me plzzzzzz answer it as soon as possible.....Prove by induction:

1/1.2 + 1/2.3 + 1/3.4 + ..... to n terms = n/(n+1)

^{5}/5+n^{3}/3+7n/15 is a natural number by using the principle of mathematical induction.using mathematical induction

prove that

7+ 77+ 777+ ..............+ 77.........7 = 7/81 (10to the power n+1 -9n-10)

1 / 1.2.3 + 1 / 2.3.4 + 1 / 3.4.5 + …….. + 1 / n (n+1) (n+2) = n (n+3) / 4 (n+1) (n+2) ? using principle of mathematical induction prove the following for all n E N ?

Prove that n

^{2}+ n is even , where n is natural number.?3.6 +6.9+9.12+....+3n(3n+3) = 3n(n+1) (n+2)

please solve.... n(n2-1) is divisible by 24,where n is an odd integer

1/2*5+ 1/5*8 + 1/8*11 +.......................+

1/(3n-1)(3n+2)

= n/6n+4

prove by PMI

prove by ibduction that the sum Sn=n

^{3}+3n^{2}+5n+3 is divisible by 3 for all nENby PMI prove , 1/1.3 + 1/3.5+ 1/5.7+.......... 1/(2n-1)(2n+1) = n/(2n+1)

Hi

By using principle of mathematical induction, prove that for all n element of N:

3^2n+2 - 8n - 9 is divisible by 64.

13. Prove that, 5

^{n}- 5 is divisible by 4 for all n $\in $ N.prove dt ....

3

^{4n+2}+ 5^{2n+1 is a multiple of 14}1+4+7+---------+(3n-2)=

1n(3n-1)2

in drilling worlds deepest hole it was found that the temperature T in degree celcius at x km below the surface of the earth was =30+25(x - 3), 3 less than x less than 15. at what depth will the temperature lie b/w 200

^{o}C and 300^{o}C.Prove by PMI

n(n+1)(n+5) is divisible by 6 for all n belongs to natural numbers

solve this equation :-

4k

^{3}+ 18k^{2}+ 23k + 9 =0 (step by step)11 power n+2 + 12 power 2n+1 is divisible by 133

Prove that

^{ 1}/_{2 }tan (^{x}/_{2})+^{1}/_{4}tan(^{x}/_{4})+...+(^{1}/_{2n})tan (^{x}/_{2n})=(^{1}/_{2n})cot(^{x}/_{2}^{n}) - cotx for all n (- N and 0<x<(^{pi}/_{2}).8. 9

^{(}^{k}^{ + 1)}+ 9^{(}^{k}^{ + 1)}− 8k − 8 − 9= {9

^{(}^{k}^{ + 1)}− 8k − 9} + 8 (9^{(}^{k}^{ + 1) }− 1)= 8

m+ 8 (9^{(}^{k}^{ + 1) }− 1)= 8{

m+ (9^{(}^{k}^{ + 1) }− 1)}plz explain the steps with reasonsprove by using pmi 1.3+2.32+3.33+...+n.3n= (2n-1)3n+1 + 3 / 4

Prove by induction that (2n+7)<(n+3)

^{2}is true7 divides 2

^{3n}-1Using the principle of mathematical induction prove that

3

^{n}$\text{\u2a7e}$ 2^{n}using mathematical induction prove that

n(n+1)(n+2) is divisible by 6.what is principle of mathamatical induction

^{2}-1) is divisible by 24 where n is an odd number greater than 2.7

^{n}-3^{n}is a divisible by 4.please solve this

x

^{n}-y^{n }is divisible by x-y

PROVE BY M.I (41)

^{n}-(14) is multiple of 27Prove the following by using the principle of mathematical induction for all

(2

n+7) < (n+ 3)^{2 }can any pls explain this question in detail??? cuz i can't get it

prove that 2

^{n}is greater than n for all positive integers n.this is example 2 from the ncert maths text book.

plzz...answer soon....i dint get the last step.