Example: 4,7,10,13,.....
The common difference is 3 and
is found by subtracting any term from the one after it.
To find the nth term, use the formula t_{n} = t_{1} + (n - 1)d where t_{1} is the first term, n is the number of the term being found, and d is the common difference.
Problem: Find a formula for the sequence 4,7,10,13,... and sketch the graph of the sequence. Then find the slope of the line containing the sequence.
Answer: t_{n} = 4 + (n - 1)(3)
= 4 + 3n - 3
= 3n + 1
To sketch the graph, plot some points such as (1,4), (2,7), (3,10),...
where x is the
number of the term being graphed and y is its value.
The graph is a set of ordered pairs,
not a line. The points are linear in nature, but they are not connected.
The slope of the line containing the sequence is found the same way the
slope of any line
is found. Select two points and find y_{2}
- y_{1.}
x_{2} - x_{1}
S = 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 + 28 + 31 + 34 + 37 + 40
Also, S = 40 + 37 + 34 + 31 + 28 + 25 + 22 + 19 + 16 + 13 + 10 + 7 + 4
Now add the two sequences together:
2S = 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44
2S = 13(44)
2S = 572
S = 286
This is the same as using the formula S_{n}
= n (t_{1} + t_{n})
or
S_{n} = n [2t_{1} + (n - 1)d]
2
2
The second formula is obtained by substituting the formula for
t_{n}into
the first one.
Now, substituting n = 13, t_{1} = 4, and d = 3, S will turn out to be 286.
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