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Patterns
To get a growth pattern you can look at the Arithmetic sequence of the numbers
and look at the difference-constant change
T1,T2, T3, T4
Eg. 4, 7, 10, 13
Diff=+3 +3 +3
get your T1 (1st term)or a= 4, get your difference =+3
so far Function is 3n
Then look at 3(1),3(2),3(3)
3, 6,
9
The difference between the two lines is +1.(4-3)
Therefore the function must be Tn =3n+1 if you replace n with 1,2,3,4 you get
the answers above
Or in the log tables under Sequences and Series you have to look for arithmetic sequence
𝑻𝒏 = 𝒂 + (𝒏 − 𝟏)𝒅
General term
a= 1st term, d= difference , n= number
T1, T2, T3,
Eg. 7, 12, 17
a=7
Diff=+5,+5
d=5
Tn = 7+(n-1)5
sub in a and d
Tn = 7+5n-5
Tn
=5n+2
check by subbing in n=1,2,3 do you get pattern above
You can find the 50th term by replacing n=50
Tn = 5(50)+2
=250+2
= 252
If you want to know which term gives an answer of 302
Tn=a+(n-1)d
302=7+(n-1)5
302=7+5n-5
302=5n+2
5n=302-2
5n=300 n=60 so on 60th term you get answer 302
Sum of the Arithmetic Series is got by adding all the numbers together
1+4+7+10
Sn = n [2a+(n-1)d]
2
a=1, d=3
Sn = n [2(1)+(n-1)3)
2
Sn = n [2+3n-3]
2
Sn = n [3n-1]
2
You can get what 16 terms are added up just replace n=16.
Sn = 16[3(16)-1]
2
Sn = 8 [48-1]
Sn =376
If given the last result and want to find the sum?
Eg. 2+5+8+….+65
Tn=65 is the last result so replace into the general equation a=2,d=3
Tn=a+(n-1)d
65=2+(n-1)3
65=2+3n-3
3n=66
Sn= 22[2(2)+(22-1)3]
2
Sn =22[4+21(3)]
2
Sn =11[67]
Sn =737
n=22