13 a Show that the sum S, of the first n terms of 7 + 14 + 28 + · . · is S, b For what value of n is S, equal to 1785? c Show that T, = 7 × 2"-1, and find how many terms are less than 70000. d Use trial-and-error to find the first sum S, that is greater than 70000. e Prove that the sum S, of the first n terms is always 7 less than the (n + 1)th term. 7(2" - 1).

Q: 1. If the sum of n terms of an A.P. is 2n + 3n^2. Find the nth term.

A: Note: according to our guidelines we can answer only first question 1) If the sum of n terms of an…

Q: if 3 does not divide n then 3 divides 2n^2+1

A: Let us Assumue 3 divides (2n2+1) n2+1=3λ Now  2n2+1=n2+1+1⇒(2n2+1)+(-λ)3=1⇒ gcd(2n2+1,3)=1…

Q: 7 8. The sum of two consecutive even integers is of their product. Find these integers. 24

A: Topic- basic maths

Q: For how many three digit numbers (100 to 999) is the sum of the digits even? (For example, 343 has…

A: Given As sum of two odd number is even and sum of any even number is even so for a three digit…

Q: solve 5.3 (using k(pi-k)/2) and setting k =pi/2 or by your own methods if you have easier ones

A: It is known to us that

Q: Give a formula for the nth positive even integer: Here is a theorem: 1+ 2+3 + ... + n п(n+1) n = Use…

A:

Q: If Sn denotes the sum of the first n terms of an AP prove that S30 = 3 (S20-S10)

A:

Q: person can encode 1000 words in every hour of typing job.which of the number n of hours that the…

A: Number of words the person can encode per hour=1000

Q: 2 Look at the following picture. 000 000 1st term 2nd term 3rd term Prove that the number of…

A: Given:

Q: 2 3 n a. If n E N, then + + + •+ 3! 4! (n+1)! (Hint: 3! (reads 3 factorial) = 3 x 2 x 1. n! = nx…

A: Induction theory: Let the given statement is P(n). Step-1: We need to check P(n) is true for n=1.(or…

Q: A sequence {a^n} of real numbers is defined recursively by a_1=2 for n ≥ 2,…

A:

Q: 1-Fd the prob. that 3(15) and 3(6s)ocaure in 8ix digit sequence. 2- Find the prob. that at least…

A: Here use no of possible arrangements of sequence and which digit occur how many time as favorable…

Q: 16. How many terms are needed to approximate 2(-1)"+1 (÷) with an error of less than 0.001?

A: The given alternating series is: ∑n=1∞(-1)n1n4 Let, an=1n4

Q: Determine if it is true or false: 1. For an = n(4n - 5)/ 2, the first term in the sequence is 1.…

A: We  are given with multiple number of question and our guideline is solve first question only.

Q: 4. Let, a, = 1, az = 2, b, = 0, b2 = 1, for n 2 3, an = 2an-1+ bn-1 for n 22 bn = bn-1+ an-1 express…

A:

Q: Can you assist  me in  verifing    that    for    all    n ≥ 1,    the    sum    of    the    first…

A: Here, we have to verify that ∀n≥1, the sum of the squares of the first 2n positive integers is given…

Q: Given that P(n) is the equation 1+3+5+7+···+(2n−1) = n2, where n is an integer such that n ≥ 1, we…

A: Principal of mathematical induction

Q: For how many positive integers 'n', n3-8n2 + 20n - 13 is a prime?

A: Introduction :

Q: 3(3 + 1) n = 3: Show that 1 + 2 + 3 =

A: To show that 1 + 2 + 3 + ...  + n = n(n + 1)2 is true for the given value of n.   Also, to show…

Q: If {an} is arithmetic, {bn} is geometric where r> 0. Given a = b1 = 2, b2 = a2, b3 = a2 + 4 Find an…

A: Here, given that an is an arithmetic sequence and bn is geometric sequence. And a1=b1=2. So,…

Q: 5. Given >U, = 3n(n +1) calculate the sum r=1 of ten terms starting from the eighth term.

A: Solution: Given that

Q: 3 divides n^3 + 2n, whenever n is a positive integer. Solve using mathematical induction

A:

Q: 1. In the number 909,507, how many times greater is the 9 in the thousands place than in the ones…

A:

Q: 37! D- Given that N= (191)(15) is an integer: i- Does the prime 11 divide N? What about the prime…

A: N = 37!19! 15!37! consist numbers that are divided by 11→11, 22 , 33→319! consist numbers that are…

Q: = 5k + 3 and b, = (k - 1)° + 4k + 4 for every integer k z 0. Let ak What are the first five terms…

A:

Q: 7 45 and fourth term of (3x -) 6. Find the product of expressed 2 as P.

A: The complete solutions are given below

Q: . Which term in the expansion of (x^2– 1/x)^9 is free of x? a. 5th b. 6th c. 7th d. 8th

A:

Q: 9. (a) Show that if n is an integer, then: -o3* (") = 4". %3D (b) Let n be a positive integer. Show…

A:

Q: 2. Prove that n²>2n for n >2

A: Note that, the given statement P(n): n2>2n for n≥2 has little correction because if n=2  then…

Q: 8. If n+1, 2n-1 and n+5 are in AP. Find the value of n

A:

Q: Let N=20. Show using the Sieve of Eratosthenes how to find all prime numbers between 10N and 10N+100…

A:

Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

A:

Q: For n ≥ 2, let an=5n/n2-2 Determine for which n, |an|<1

A: Use the given inequality for an and find the quadratic equation for n and solve

Q: Show that the sum of the squares of the first n natural numbers is given by,S=(n(n+1)(2n+1))/6

A: We have to show that the sum of the squares of the first n natural numbers is given by S=nn+12n+16…

Q: [from n=4 to infinity] E (2/3)3n-1 (-4/7)2n+3 *that E just means that sum symbol.

A: Let\'s say the sum of the series is S. Hence S will be as shown on the white board. Please note the…

Q: Determine how many terms of (-1)*+12k k! must be summed to be sure that the remainder term k=1 is…

A:

Q: For n≥2, let an= 5n/n2-2. Determine for which n, |an|<1.

A: Use the inequality on and find the inequality for n

Q: (c) How many strings of four decimal digits have exactly three digits that are 9? (d) Write down the…

A: We have to find the coefficient.

Q: 1. Prove that the sum of the consecutive cubes is given by this formula: n²(n+1) 4 13 + 23 + 33 +..+…

A: Given; 1) 13+23+33+...+n3=n2(n+1)2/4 Repost the next questions.

Q: 4. Theoretically, what is the highest possible value for the slugging average? Could a player ever…

A: Given, the baseball example.     We have to find the highest value for slugging average?

Q: The sum of the digits in the number 56 is 11. (5+6=11) a) Show that the sum of the digits of the…

A: a) Sum of the digits from 15 to 18 1+5=6 1+6=7 1+7=8 1+8=9 Now 6+7+8+9=30

Q: 4 (a) Write out the multiplication table for Z6. How many times does [0]6 appear in each row of the…

A:

Q: Verify that for all n ≥ 1, the sum of the squares of the first 2n positive integers is given by the…

A: Verify that for all n ≥ 1, the sum of the squares of the first 2n positive integers is given bythe…

Q: 5) Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4.…

A:

Q: How many numbers have digits which are all ”1”s and ”2”s, and thedigits add up to 8? A few such…

A:

Q: 10. Prove that the sum of n squares can be found as follows 12 + 22 + 3°+...+n² n(n + 1)(2n + 1) 6.

A:

Q: Use Gauss’s trick to determine H(n) = 1 + 5 + 9 + 13 + … + (4n – 7) + (4n – 3) + (4n + 1) I'm…

A: Explained below

Q: 2n2 – 4n +1 is even → n is odd

A:

Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.