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Syllabus

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

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

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

2^5n>3^3n for n belongs to natural no.

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

13. Prove that, 5

^{n}- 5 is divisible by 4 for all n $\in $ N.Prove that 2.7n + 3.5n - 5 is divisible by 24 for all n belongs to N. [ please explain with steps]

1 sqare + 2 square + 3 square +.............+n square >n cube +3 . prove it by mathematical induction.

Prove that x

^{2n}-y^{2n}is divisible by x+y?9n - 8n -1 is a multiple of 8

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

By Principle of Mathematical Induction, prove that:

12

^{n}+ 2.5^{n-1}is divisible by 7.

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.

x

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

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

13. Find the square root of $4+6\sqrt{-5}$

Using PMI Prove that 1^2 + 2^2 + ......... + n^2 n^3/3

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.prove dt ....

3

^{4n+2}+ 5^{2n+1 is a multiple of 14}find the equation of parabola whose focus is (1,1) and tangent at the vertex is x + y = 1

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

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

= n/6n+4

prove by PMI

Prove that 11

^{n+2 }+ 12^{2n+1 }is divisible by 133 for all n belongs to N .Prove: 5

^{2n}-1 is divisible by 24 for all n Nfor every positive integer n.prove that 7

^{n}-3^{n}is divisible by 4Using PMI, prove that

5

^{2n+2}-24n-25 is divisible by 576 for n belongs to N.12

^{n}+ 2.5^{n-1}is divisible by 7.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

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.Prove the following by the PMI :

n7/ 7 + n5 / 5 + n3 / 3 + n2/ 2 - 37/210n is a positive integer for all n belongs to N.

how to split a cubic polynomial like k

^{3}+6k^{2}+9k+4??bY PMI prove n(n+1)(2n+1) is divisible by 6

n

^{7}/7 + n^{5}/5 + 2n^{3}/3 - n/105 is an integer , prove by PMI.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.^{n-1}x) = sin 2^{x}a/ 2^{x}sin ausing mathematical induction

prove that

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

11 power n+2 + 12 power 2n+1 is divisible by 133

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 by ibduction that the sum Sn=n

^{3}+3n^{2}+5n+3 is divisible by 3 for all nENProve that n

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

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

For all positive integersuch that

Hi

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

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

^{2n}+2^{2n-3}3^{n-1 }is divisible by 251+4+7+---------+(3n-2)=

1n(3n-1)2

Prove by PMI

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

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.way we take 8m in example no 1 ????????

what is LUCAS SEQUENCE?

a(n) = a(n-1)+ a(n-2), n>2 ;a1=1 , a2= 3 , prove a(n)<or = to (1.75)raise to n

_{===aa}Prove by induction that (2n+7)<(n+3)

^{2}is trueSuppose there is a given statement P(n) involving the natural numbernsuch thatThe statement is true forn= 1, i.e., P(1) is true

What "P" MEANS and How P(1) IS true??????????Please explain I don't got it frm study material.............7 divides 2

^{3n}-1using PMI prove that 3 to the power (2n-1) is divisible by 8 where n belongs to N.

plz reply

using mathematical induction prove that

n(n+1)(n+2) is divisible by 6.1. 1

^{2}+3^{2}+5^{2}++(2n-1)^{2}=2-1)what is principle of mathamatical induction

Prove that, x

^{2n}- y^{2n}is divisible by x+y.7

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

^{2n}+ 16n -1 is divisible by 64 Prove by mathematical inductionPROVE 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.

$U\mathrm{sin}gtheprincipalofmathematicalinductionprovethat\phantom{\rule{0ex}{0ex}}\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{{2}^{n}}=1-\frac{1}{{2}^{n}}\phantom{\rule{0ex}{0ex}}foralln\in N$