Advertisements
Advertisements
प्रश्न
By the principle of mathematical induction, prove the following:
1 + 4 + 7 + ……. + (3n – 2) = `("n"(3"n" - 1))/2` for all n ∈ N.
Advertisements
उत्तर
Let P(n) : 1 + 4 + 7 + ……. + (3n – 2) = `("n"(3"n" - 1))/2`
Put n = 1,
LHS = 1
RHS = `(1(3 - 1))/2 = 1`
∴ P(1) is true.
Assume P(k) is true for n = k
P(k): 1 + 4 + 7 + ……. + (3k – 2) = `(k(3k - 1))/2`
To prove P(k + 1) is true, i.e., to prove
1 + 4 + 7 + ……. + (3k – 2) + (3(k + 1) – 2) = `((k + 1)(3(k + 1) - 1))/2`
1 + 4 + 7 + ……. + (3k – 2) + (3k + 3 – 2) = `((k + 1)(3k + 2))/2`
1 + 4 + 7 + …… + (3k + 1) = `((k + 1)(3k + 2))/2`
1 + 4 + 7 + …… + (3k – 2) + (3k + 1) = `(k(3k - 1))/2 + (3k + 1)`
`= (k(3k - 1) + 2(3k + 1))/2`
`= (3k^2 - k + 6k + 2)/2`
`= (3k^2 + 5k + 2)/2`
`= ((k + 1)(3k + 2))/2`
∴ P(k + 1) is true whenever P(k) is true.
∴ By the Principle of Mathematical Induction, P(n) is true for all n ∈ N.
APPEARS IN
संबंधित प्रश्न
By the principle of mathematical induction, prove the following:
1.2 + 2.3 + 3.4 + … + n(n + 1) = `(n(n + 1)(n + 2))/3` for all n ∈ N.
By the principle of mathematical induction, prove the following:
4 + 8 + 12 + ……. + 4n = 2n(n + 1), for all n ∈ N.
The term containing x3 in the expansion of (x – 2y)7 is:
Using the Mathematical induction, show that for any natural number n ≥ 2,
`1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 +2 + 3 + 4) + .... + 1/(1 + 2 + 3 + ... + "n") = ("n" - 1)/("n" + 1)`
Using the Mathematical induction, show that for any natural number n,
`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 the Mathematical induction, show that for any natural number n,
`1/(2.5) + 1/(5.8) + 1/(8.11) + ... + 1/((3"n" - 1)(3"n" + 2)) = "n"/(6"n" + 4)`
Using the Mathematical induction, show that for any natural number n, x2n − y2n is divisible by x + y
By the principle of Mathematical induction, prove that, for n ≥ 1
`1^2 + 2^2 + 3^2 + ... + "n"^2 > "n"^2/3`
Choose the correct alternative:
In 3 fingers, the number of ways four rings can be worn is · · · · · · · · · ways
Choose the correct alternative:
Everybody in a room shakes hands with everybody else. The total number of shake hands is 66. The number of persons in the room is ______
