document.write( "Question 542370: find the sum of the following arithmetic sequence
\n" ); document.write( "1+5+9+...+405 \n" ); document.write( "

Algebra.Com's Answer #354560 by KMST(5328) $\"\"$ $\"About$

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There is a formula to calculate the sum of an arithmetic sequence, but most of the time it's easier (at least for me) to reason in the same way that lead to finding that formula.
\n" ); document.write( "If you had to add the numbers from 1 to 99, you could add them twice pairing the terms of the sums head to tail like this:
\n" ); document.write( "(1+2+3+4+...+96+97+98+99)+(99+98+97+96+...+4+3+2+1)= (1+99)+(2+98)+(3+97)+...+(97+3)+(98+2)+(99+1)= 100+100+100+...+100+100+100=99*100=9900
\n" ); document.write( "and then divide by two. So you could say that the sum of an arithmetic sequence is the number of terms, times the sum of the first and last terms, divided by two.
\n" ); document.write( "In this case there are 102 terms. (The first is 1, and the 102nd one is 405. So, 1+5+9+...+405=102*(405+1)/2=102*406/2=20706
\n" ); document.write( "But if you like formulas, you have $\"a%281%29=1\"$ , $\"n=102%29%29%29%2C++%7B%7B%7Bd=4\"$ ,
\n" ); document.write( " $\"a%28n%29=a%281%29%2B%28n-1%29d\"$ , $\"a%28102%29=1%2B%28102-1%29%2A4=1%2B%28101%29%2A4=1%2B404=405\"$
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