If n th term of a sequence is 2an+b, where a, b are constants, is this sequence an a.p.?
term(n) = a + (n-1)d
for yours:
term(n) = 2an+b
term(n-1) = 2a(n-1) + b = 2an - 2a + b
term(n+1) = 2a(n+1) + b = 2an + 2a + b
common difference = term(n) - term(n-1)
= 2an + b - (2an - 2a + b)
= 2a
or common difference = term(n+1) - term(n)
= 2an+2a + b - (2an + b)
= 2a
yes, it is
Well, if the sequence is following the pattern of 2an+b, then it might seem like an arithmetic progression (a.p.) at first. However, let me tell you a secret: appearances can be deceptive, just like my clown makeup! 🤡
To determine whether this sequence is truly an arithmetic progression, we need to check if the difference between consecutive terms is constant. In other words, we need to check if (2a(n+1)+b) - (2an+b) is always the same.
Expanding that expression, we get 2a(n+1) + b - 2an - b, which simplifies to 2an + 2a + b - 2an - b. The "b" terms cancel out, and we are left with 2a(n+1) - 2an. Simplifying further, we get 2a.
So, the difference between consecutive terms is 2a, which means it is constant! 🎉 Hence, the sequence is indeed an arithmetic progression. It seems the clown knows a thing or two about sequences too! 🤡
Yes, the given sequence is an arithmetic progression (a.p.).
In an arithmetic progression, the difference between consecutive terms is always constant. Let's analyze the given sequence to check if it satisfies this condition.
The general form of an arithmetic progression is given by Tn = a + (n-1)d, where Tn represents the n-th term, a is the first term, n is the position of the term, and d is the common difference.
Comparing this form with the given sequence, Tn = 2an + b, we can rewrite it as Tn = an + (an + b).
From this, we can see that the value of the first term is a, and the common difference between consecutive terms is (an + b) - (a) = an.
Since the common difference is independent of the position n, this implies that the given sequence is indeed an arithmetic progression (a.p.).
To determine if a sequence is an arithmetic progression (a.p.), we need to check if the common difference between consecutive terms is the same.
In the given sequence, the n-th term is represented as 2an + b. Let's find the difference between consecutive terms:
The (n+1)-th term:
2a(n+1) + b
Difference between consecutive terms (d):
(2a(n+1) + b) - (2an + b) = 2an + 2a + b - 2an - b = 2a
From this calculation, we can observe that the difference between consecutive terms (d) is 2a, which is a constant value and does not depend on the value of n.
Hence, we can conclude that the given sequence is indeed an arithmetic progression (a.p.) since it has a constant difference between consecutive terms.