+130536 So we would have
4 , 7 , 10 , 13 , 16 , 19 , etc. and the final arrangement (term) is given by
an = a1 + d(n-1) where an is the final term, a1 is the first term, d is the arithmetic difference between successive terms, and n is the number of terms......so we have
an = 4 + 3(250 - 1) = 4 + 3(249) = 751 toothpicks
+130536 So we would have
4 , 7 , 10 , 13 , 16 , 19 , etc. and the final arrangement (term) is given by
an = a1 + d(n-1) where an is the final term, a1 is the first term, d is the arithmetic difference between successive terms, and n is the number of terms......so we have
an = 4 + 3(250 - 1) = 4 + 3(249) = 751 toothpicks
![The first three stages of a pattern are shown below, in which each line segment represents a toothpick. If the pattern continues such that at each successive stage, three toothpicks are added to the previous arrangement, how many toothpicks are necessary to create the arrangement for the 250th stage? [asy] size(150); defaultpen(linewidth(0.7)); void drawSquare(pair A){ draw((A.x + 0.1,A.y)--(A.x + 0.9,A.y)); draw((A.x,A.y + 0.1)--(A.x,A.y + 0.9)); draw((A.x + 1,A.y + 0.1)--(A.x + 1,A.y + 0.9)); draw((A.x + 0.1,A.y + 1)--(A.x + 0.9,A.y + 1)); } int k = 0; for(int i = 1; i <= 3; ++i){ for(int j = 0; j < i; ++j){ drawSquare((k,0)); ++k; } draw((k+0.1,0.5)--(k+0.9,0.5),EndArrow); ++k; } label("$\cdots$",(k,0.5)); [/asy]](/api/ssl-img-proxy?src=https%3A%2F%2Fdata.artofproblemsolving.com%2Faops20%2Flatex%2Fimages%2F89ec76d7d4f948a98e0b35ad551447f2c6d5b889.png)
