Pascal triangle contains values as [tex]^nC_r[/tex] ; n being row number, and r is index in row, both 0 indexing. The second term in expansion of [tex](x+2)^4[/tex] is [tex]8x^3[/tex]
What is Pascal triangle?
Pascal triangles contains coefficient of [tex](x+1)^n[/tex] 's expanded form's terms.
For first row, we have:
[tex](x+1)^0 = 1[/tex]
Thus, 1 is the value in the first row of Pascal triangle.
Thus, for second row, we have:
[tex](x+1)^1 = x + 1[/tex]
The variable x has coefficient 1, and that 1 itself, so second row of Pascal triangle has 1 , 1 as values.
For third row, we have:
[tex](x+1)^2 = x^2 + 2x + 1[/tex]
Thus, 1, 2, 1 as values in third row. And so on.
For nth row, and its rth term(all counted from 0-indexing(that means, counting of index starts from 0), then that term in Pascal triangle is [tex]^nC_r[/tex]
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
.........
The rth indexed term with nth row is termed as [tex]^nC_rx^{n-r}y^r[/tex] for expansion of [tex](x+y)^n[/tex]
For the given condition, we've got [tex](x+2)^4[/tex]
We have n = 4, and r = 2 - 1 = 1(as second term needed, and second term is 1 indexed in 0 indexing).
Since y = 2, thus, the second term of [tex](x+2)^4[/tex] is [tex]^4C_1x^32^1 = 8x^3[/tex]
Thus,
The second term in expansion of [tex](x+2)^4[/tex] is [tex]8x^3[/tex]
Learn more about Pascal triangle here:
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