- #1
annoymage
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Homework Statement
find the formula for
1,3,6,10,15,21,...
Homework Equations
n/a
The Attempt at a Solution
i only can find n>=3
Tn = 3 + [tex]\sum[/tex] i ; i=3 to n
help T_T
But 3= 1+ 2 and 1= 0+ 1 so you can say it is [itex]T_n= \sum_{i= 1}^n i[/itex] That's a well known sum with a well known formula. Look up "triangular numbers".annoymage said:Homework Statement
find the formula for
1,3,6,10,15,21,...
Homework Equations
n/a
The Attempt at a Solution
i only can find n>=3
Tn = 3 + [tex]\sum[/tex] i ; i=3 to n
help T_T
The formula for this sequence is n(n+1)/2, where n represents the position of the term in the sequence. For example, the first term in the sequence is 1, so n=1. Plugging this into the formula, we get 1(1+1)/2 = 2/2 = 1. The second term is 3, so n=2. Plugging this into the formula, we get 2(2+1)/2 = 6/2 = 3, and so on.
Yes, the formula can also be written as (n^2 + n)/2. This is known as the triangular number formula.
Finding a formula for a sequence allows us to easily predict and calculate any term in the sequence without having to write out each term. It also helps us understand the pattern and behavior of the sequence.
Yes, there are many common sequences with similar formulas, such as the arithmetic sequence (where the difference between consecutive terms is constant) and the geometric sequence (where the ratio between consecutive terms is constant).
Yes, the formula for the sum of an arithmetic sequence (Sn) is n/2(2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference. In this case, a=1 and d=1, so the formula simplifies to n/2(n+1), which is equivalent to the formula for the nth term in this sequence.