



 
Hi Fredy. You need to look at the sequence 2, 6, 12, 20, 30, 42 and figure out an algebraic pattern for the i'th term. Two ways of doing this. One is intuitive (just playing with numbers in your head and "seeing" the answer). I think I see it. But if I didn't "see" this, how could I work it out mathematically? I'll show you how: The first differences (differences between adjacent terms) are 4, 6, 8, 10, 12. The zeroth term, if we project backwards, would be 2 less than the first term. That's a zero. So t_{0} = 0. Thus So that simplifies it to just t_{i} = Ai^{2} + Bi, which factors to i(Ai + B). That means we can write out the terms in order We're getting closer.... So the equation we have is this: t_{i} = 1i^{2} + 1B + 0. In factored form, that is i(i + 1). This is the same as what I saw "intuitively" above, but it is mathematically proven. In sigma notation, you just need to know now what i to start with (since you want 2 for the first term, that is 1 x 2, so i = 1) and what i to end on (last number is 42, so that's 6 x 7, so the last i is 6). Hope this helps,  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 