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\n" ); document.write( "a.
\n" ); document.write( "The common difference d in the sequence is 9-4=5.
\n" ); document.write( "a(2) = a(1)+1(d) = a(1)+1(5) = 4+5 = 9 [the second term is the first term, plus the common difference 1 time]
\n" ); document.write( "a(3) = a(1)+2(d) = a(1)+2(5) = 4+10 = 14 [the third term is the first term, plus the common difference 2 times]
\n" ); document.write( "...
\n" ); document.write( "a(n) = a(1)+(n-1)(d) = a(1)+(n-1)(5) = 4+10 = 14 [the n-th term is the first term, plus the common difference (n-1) times]
\n" ); document.write( "Your textbook probably shows a formula something like \"a(n) = a(1)+(n-1)(d)\" for the n-th term in an arithmetic sequence. But don't just memorize the formula; look at what it says and see that it is just common sense based on the definition of an arithmetic sequence.
\n" ); document.write( "So the 25th term in your sequence is the first term, plus the common difference 24 times: 4 + 24(5) = 4+120 = 124.
\n" ); document.write( "Answer for part a: the 25th term is 124.
\n" ); document.write( "b.
\n" ); document.write( "The formula you probably have for the sum of the first n terms of an arithmetic sequence is probably an ugly one; I suggest you don't try to memorize it. Instead, think of finding the sum of the terms of an arithmetic sequence like this....
\n" ); document.write( "(1) the sum of the terms is the number of terms, multiplied by the average of all the terms; and
\n" ); document.write( "(2) because the terms of an arithmetic sequence are equally spaced, the average of all the terms is just the average of the first and last terms.
\n" ); document.write( "So to find the sum of the terms of an arithmetic sequence we only need to know the first and last terms and the number of terms.
\n" ); document.write( "4, 7, 10, 13, ... out to 60 terms.
\n" ); document.write( "The first term is 4; the common difference is 3; so the 60th term is the first term plus the common difference 59 times: 4+59(3) = 4+177 = 181.
\n" ); document.write( "The average of the first and last terms -- and therefore the average of all the terms, is (4+181)/2 = 185/2.
\n" ); document.write( "The sum of the first 60 terms is the number of terms, multiplied by the average of all the terms:
\n" ); document.write( "
\n" ); document.write( "Answer for part b: the sum of the first 60 terms is 5550. \n" ); document.write( "