Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 11 TN Board chapter 5 - Binomial Theorem, Sequences and Series [Latest edition]

Expand `(2x^2 - 3/x)^3`

Expand `(2x^2 -3sqrt(1 - x^2))^4 + (2x^2 + 3sqrt(1 - x^2))^4`

Compute 102⁴ 

Compute 99⁴ 

Compute 9⁷ 

Using binomial theorem, indicate which of the following two number is larger: `(1.01)^(1000000)`, 10

Find the coefficient of x¹⁵ in `(x^2 + 1/x^3)^10`

Find the coefficient of x² and the coefficient of x⁶ in `(x^2 -1/x^3)^6` 

Find the coefficient of x⁴ in the expansion `(1 + x^3)^50 (x^2 + 1/x)^5`

Find the constant term of `(2x^3 - 1/(3x^2))^5`

Find the last two digits of the number 3⁶⁰⁰ 

If n is a positive integer, using Binomial theorem, show that, 9ⁿ⁺¹ − 8n − 9 is always divisible by 64

If n is an odd positive integer, prove that the coefficients of the middle terms in the expansion of (x + y)ⁿ are equal

If n is a positive integer and r is a non-negative integer, prove that the coefficients of x^r and x^n−r in the expansion of (1 + x)ⁿ are equal

If a and b are distinct integers, prove that a − b is a factor of aⁿ − bⁿ, whenever n is a positive integer. [Hint: write aⁿ = (a − b + b)ⁿ and expaand]

In the binomial expansion of (a + b)ⁿ, if the coefficients of the 4^th and 13^th terms are equal then, find n

If the binomial coefficients of three consecutive terms in the expansion of (a + x)ⁿ are in the ratio 1 : 7 : 42, then find n

In the binomial expansion of (1 + x)ⁿ, the coefficients of the 5^th, 6^th and 7^th terms are in AP. Find all values of n

Prove that `"C"_0^2 + "C"_1^2 + "C"_2^2 + ... + "C"_"n"^2 = (2"n"!)/("n"!)^2`

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them

`1/(2^("n"+ 1))`

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them

`(("n" + 1)("n" + 2))/(("n" + 3)("n" + 4))`

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them

`4 (1/2)^"n"`

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them

`(- 1)^"n"/"n"`

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them

`(2"n" + 3)/(3"n" + 4)`

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them

2018

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them

`(3"n" - 2)/(3^("n" - 1))`

Write the first 6 terms of the sequences whose n^th term a_n is given below

a_n = `{{:("n" + 1,  "if"  "n is odd"),("n",  "if"  "n is even"):}`

Write the first 6 terms of the sequences whose n^th term a_n is given below

a_n = `{{:(1,  "if n" = 1),(2,  "if n" = 2),("a"_("n" - 1) + "a"_("n" - 2),  "if n" > 2):}}` 

Write the first 6 terms of the sequences whose n^th term a_n is given below

a_n = `{{:("n",  "if n is"  1","  2  "or"  3),("a"^("n" - 1) + "a"_("n" - 2) + "a"_("n" - 3), "if n" > 3):}`

Write the n^th term of the following sequences.
2, 2, 4, 4, 6, 6, . . .

Write the n^th term of the following sequences.
`1/2, 2/3, 3/4, 4/5, 5/6, ...`

Write the n^th term of the following sequences.
`1/2, 3/4, 5/6, 7/8, 9/10, ...`

Write the n^th term of the following sequences.
6, 10, 4, 12, 2, 14, 0, 16, −2, . . .

The product of three increasing numbers in GP is 5832. If we add 6 to the second number and 9 to the third number, then resulting numbers form an AP. Find the numbers in GP

Write the n^th term of the sequence `3/(1^2 2^2), 5/(2^2 3^2), 7/(3^2 4^2), ...` as a difference of two terms

If t_k is the k^th term of a G.P., then show that t_{n – k}, t_n, t_{n + k} also form a GP for any positive integer k

If a, b, c are in geometric progression, and if `"a"^(1/x) = "b"^(1/y) = "C"^(1/z)`, then prove that x, y, z are in arithmetic progression

The AM of two numbers exceeds their GM by 10 and HM by 16. Find the numbers

If the roots of the equation (q – r)x² + (r – p)x + p – q = 0 are equal, then show that p, q and r are in AP

If a , b , c are respectively the p^th, q^th and r^th terms of a G . P show that (q – r) log a + (r – p) log b + (p – q) log c = 0

Find the sum of the first 20-terms of the arithmetic progression having the sum of first 10 terms as 52 and the sum of the first 15 terms as 77

Find the sum up to the 17^th term of the series `1^3/1 + (1^3 + 2^3)/(1 + 3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) + ...` 

Compute the sum of first n terms of the following series:
8 + 88 + 888 + 8888 + ...

Compute the sum of first n terms of the following series:
6 + 66 + 666 + 6666 + ...

Compute the sum of first n terms of 1 + (1 + 4) + (1 + 4 + 4²) + (1 + 4 + 4² + 4³) + ...

Find the general term and sum to n terms of the sequence `1, 4/3, 7/9, 10/27, ......`

Find the value of n, if the sum to n terms of the series `sqrt(3) + sqrt(75) + sqrt(243) + ......` is `435 sqrt(3)`

Show that the sum of (m + n)^th and (m − n)^th term of an AP. is equal to twice the m^th term

A man repays an amount of Rs.3250 by paying Rs.20 in the first month and then increases the payment by Rs.15 per month. How long will it take him to clear the amount?

In a race, 20 balls are placed in a line at intervals of 4 meters, with the first ball 24 meters away from the starting point. A contestant is required to bring the balls back to the starting place one at a time. How far would the contestant run to bring back all balls?

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2^nd hour, 4^th hour and n^th hour?

What will Rs.500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

In a certain town, a viral disease caused severe health hazards upon its people disturbing their normal life. It was found that on each day, the virus which caused the disease spread in Geometric Progression. The amount of infectious virus particle gets doubled each day, being 5 particles on the first day. Find the day when the infectious virus particles just grow over 1,50,000 units?

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid

`1/(5 + x)`

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid

`2/(3 + 4x)^2`

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid

`(5 + x^2)^(2/3)`

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid

`(x + 2) - 2/3`

Find `root(3)(10001)` approximately (two decimal places

Prove that `root(3)(x^3 + 6) - root(3)(x^3 + 3)` is approximately equal to `1/x^2` when x is sufficiently large

Prove that `sqrt((1 - x)/(1 + x))` is approximately euqal to `1 - x + x^2/2` when x is very small

Write the first 6 terms of the exponential series
e^5x 

Write the first 6 terms of the exponential series
`"e"^(-2x)`

Write the first 6 terms of the exponential series
`"e"^(1/2x)`

Write the first 4 terms of the logarithmic series
log(1 + 4x) Find the intervals on which the expansions are valid.

Write the first 4 terms of the logarithmic series
log(1 – 2x) Find the intervals on which the expansions are valid.

Write the first 4 terms of the logarithmic series
`log((1 + 3x)/(1 -3x))` Find the intervals on which the expansions are valid.

Write the first 4 terms of the logarithmic series
`log((1 - 2x)/(1 + 2x))` Find the intervals on which the expansions are valid.

If y = `x + x^2/2 + x^3/3 + x^4/4  ...`, then show that x = `y - y^2/(2!) + y^3/(3!) - y^4/(4) + ...`

If p − q is small compared to either p or q, then show `root("n")("p"/"q")` ∼ `(("n" + 1)"p" + ("n" - 1)"q")/(("n"- 1)"p" +("n" + 1)"q")`. Hence find `root(8)(15/16)`

Find the coefficient of x⁴ in the expansion `(3 - 4x + x^2)/"e"^(2x)`

Find the value of `sum_("n" = 1)^oo 1/(2"n" - 1) (1/(9^("n" - 1)) + 1/(9^(2"n"- 1)))`

Choose the correct alternative:
The value of 2 + 4 + 6 + … + 2n is

Choose the correct alternative:
The coefficient of x⁶ in (2 + 2x)¹⁰ is

Choose the correct alternative:
The coefficient of x⁸y¹² in the expansion of (2x + 3y)²⁰ is

Choose the correct alternative:
If ⁿC₁₀ > ⁿC_r for all possible r, then a value of n is

Choose the correct alternative:
If a is the arithmetic mean and g is the geometric mean of two numbers, then

Choose the correct alternative:
If (1 + x²)² (1 + x)ⁿ = a₀ + a₁x + a₂x² + …. + x^{n + 4} and if a₀, a₁, a₂ are in AP, then n is

Choose the correct alternative:
If a, 8, b are in A.P, a, 4, b are in G.P, if a, x, b are in HP then x is

Choose the correct alternative:
The sequence = `1/sqrt(3), 1/(sqrt(3) + sqrt(2)), 1/(sqrt(3) + 2sqrt(2)) ...` form an

Choose the correct alternative:
The HM of two positive numbers whose AM and GM are 16, 8 respectively is

Choose the correct alternative:
If S_n denotes the sum of n terms of an AP whose common difference is d, the value of S_n − 2S_n−1 + S_n−2 is

Choose the correct alternative:
The remainder when 38¹⁵ is divided by 13 is

Choose the correct alternative:
The n^th term of the sequence 1, 2, 4, 7, 11, …… is

Choose the correct alternative:
The sum up to n terms of the series `1/(sqrt(1)  +sqrt(3)) + 1/(sqrt(3) + sqrt(5)) + 1/(sqrt(5) + sqrt(7)) + ...` is 

Choose the correct alternative:
The n^th term of the sequence `1/2, 3/4, 7/8, 15/16, ...` is

Choose the correct alternative:
The sum up to n terms of the series `sqrt(2) + sqrt(8) + sqrt(18) + sqrt(32) + ...` is

Choose the correct alternative:
The value of the series `1/2 + 7/4 + 13/8 + 19/16 + ...` is

Choose the correct alternative:
The sum of an infinite GP is 18. If the first term is 6, the common ratio is

Choose the correct alternative:
The coefficient of x⁵ in the series e^-2x is

Choose the correct alternative:
The value of `1/(2!) + 1/(4!) + 1/(6!) + ...` is

Choose the correct alternative:
The value of `1 - 1/2(2/3) + 1/3(2/3)^2  1/4(2/3)^3 + ...` is