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What does descending in algebra mean?
Descending (in a sequence) means that a the next number is "more negative" or "closer to negative infinity" or "less positive" or "further from positive infinity" or if n is a number in a sequence and n+1 is the next number then n/n+1 > 1
What is the form of the nth term of a triangular pyramidal number sequence?
1/6 n(n+1)(n+2)
What is thenext logical number 1 2 6 42 1086?
If x(n) represents the nth number is the sequence x(n+1)=x(n)*(x(n)+1) So the next number in the sequence is 1086*(1086+1)=3263442
What is the formula to find sum of n odd numbers?
The set of odd numbers is an arithmetic sequence. Let say that the sequence has n odd numbers where the first term is a1 and the last one is n. The formula to find the sum on nth terms for an arithmetic sequence is: Sn = (n/2)(a1 + an) or Sn = (n/2)[2a1 + (n - 1)d] where d is the common difference that for odd numbers is 2. Sn = (n/2)(2a1 + 2n - 2)
What is the sum of the first 22 odd numbers?
The sum of a sequence is given by sum = n/2(2a + (n-1)d) where: n = how many a = first number of sequence d = difference between terms of sequence. For the first 22 odd numbers these are: n = 22 a = 1 d = 2 → sum = 22/2(2×1 + (22 - 1)×2)) = 22² = 484 The sum of the first n odd numbers is always n²: sum = n/2(2×1 + (n-1)2) = n/2(1 + (n-1))×2 = n(n) = n²
How do you prove that the sum of a convergent sequence divided by n will converge?
You can use the comparison test. Since the convergent sequence divided by n is less that the convergent sequence, it must converge.
Does -1n converge?
No, -1^n does not converge as it alternates between -1 and 1 for different values of n. This oscillation prevents the sequence from approaching a specific limit.
Prove that every convergent sequence is a Cauchy sequence?
The limits on an as n goes to infinity is aThen for some epsilon greater than 0, chose N such that for n>Nwe have |an-a| < epsilon.Now if m and n are > N we have |an-am|=|(am -a)-(an -a)|< or= |am -an | which is < or equal to 2 epsilor so the sequence is Cauchy.
How do you find the period of a sequence?
To find the period of a sequence, identify the smallest positive integer ( n ) such that the terms of the sequence repeat every ( n ) positions. This can be done by examining the sequence for patterns or cycles. If you notice that the sequence returns to its initial values after ( n ) terms, then ( n ) is the period. If no such ( n ) exists, the sequence is considered aperiodic.
How do you find the harmonic and geometric mean for grouped data?
The geometric-harmonic mean of grouped data can be formed as a sequence defined as g(n+1) = square root(g(n)*h(n)) and h(n+1) = (2/((1/g(n)) + (1/h(n)))). Essentially, this means both sequences will converge to the mean, which is the geometric harmonic mean.
Define a monotone sequence of real numbers?
A monotone increasing sequence {r_n | n>0} is a sequence with: n>m implies r_n >= r_m A monotone decreasing sequence {r_n | n>0} is a sequence with: n>m implies r_n <= r_m A strictly monotone increasing sequence {r_n | n>0} is a sequence with: n>m implies r_n > r_m A strictly monotone decreasing sequence {r_n | n>0} is a sequence with: n>m implies r_n < r_m Theorem. All bounded monotone sequences of real numbers have a unique limit.
What is meant by converge and diverge?
Converge means coming together or meeting at a common point, while diverge means branching off or moving apart in different directions. In mathematical terms, if a sequence or series of values approaches a specific number as they progress, they are said to converge. On the other hand, if the values in a sequence or series move further apart or do not approach a specific number, they are said to diverge.
Why is there no formula in getting the sum of a harmonic sequence?
A harmonic sequence is defined as a sequence of the form ( a_n = \frac{1}{n} ), where ( n ) is a positive integer. The sum of a harmonic series, ( \sum_{n=1}^{N} \frac{1}{n} ), diverges as ( N ) approaches infinity, meaning it grows without bound. Unlike arithmetic or geometric series, which have closed-form sums due to their consistent growth patterns, the harmonic series does not converge to a finite limit, making it impossible to express its sum with a simple formula. Thus, while there are approximations (like the use of logarithms), there is no exact formula for the sum of an infinite harmonic series.
Explain how to find the terms of Fibonacci sequence?
If the Fibonacci sequence is denoted by F(n), where n is the first term in the sequence then the following equation obtains for n = 0.
Is a geometric progression a quadratic sequence?
A geometric sequence is : a•r^n while a quadratic sequence is a• n^2 + b•n + c So the answer is no, unless we are talking about an infinite sequence of zeros which strictly speaking is both a geometric and a quadratic sequence.
Does 1 over n converge?
No. ∑(1/n) diverges. It is the special infinite series known as the "harmonic series."
What is convergence in numeric math?
A sequence of numbers, xn (where n = 1, 2, 3, ...), is said to converge to a limit L if, given any positive e, however small, it is possible to find an integer k such that |xn - L| < e for all n > k.In other words, after a certain point (k) all terms of the sequence are closer to the limit (L) than any tiny number.
