anonymous
  • anonymous
1,3,6,10,15 whats the 15th and 30th term
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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campbell_st
  • campbell_st
looks like the set of triangular numbers \[a_{n} = a_{n -1} + n\]
anonymous
  • anonymous
@campbell_st what? im so confused..
campbell_st
  • campbell_st
the sequence is well know.... like the set of square numbers... you can draw a triangle with numbers |dw:1377719816292:dw| the sequence is increasing by the counting numbers1, 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10... so the 6th number is 15 + 6 = 21 7th = 21 + 7 = 28 so the formula for a term is \[a_{n} = \frac{n(n + 1)}{2}\] check to see if the formula fits... if it does then its easy to find the 15th and 30th numbers in the sequence

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ybarrap
  • ybarrap
Generally, to derive an sequence, start with the 1st and work your way up to see the pattern: $$ a_1=1\\ a_2=a_1+2=3\\ a_3=a_2+3=6\\ a_4=a_3+4=10\\ \cdots\\ a_n=a_{n-1}+n\\ $$ To derive the formula, bring variables to one side: $$ a_1-0=1\\ a_2-a_1=2\\ a_3-a_2=3\\ a_4-a_3=4\\ \cdots\\ a_n-a_{n-1}=n\\ $$ Sum both sides of the equation, everything on the left cancels except \(a_n\) $$ a_n=1+2+\cdots+n={n(n+1)\over2} $$ The formula \({\large n(n+1)\over2}\) results because \(1+2+\cdots+n\) is an arithmetic series: http://en.wikipedia.org/wiki/Arithmetic_series . This just happens to also be the value of our arithmetic \( sequence,~a_n\) Let me know if you have any questions.
campbell_st
  • campbell_st
but the difference between terms isn't a constant... so its not an arithmetic sequence
ybarrap
  • ybarrap
The difference between terms isn't a constant, but the difference between the difference is a constant (i.e. 1,2,..,n).

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