Find the sum - r-1 n r r-1 r-2 r-3

∑_r=1^nr(r+1)(r+2)(r+3) ∑_r=1^n(r^2+r)(r^2+5r+6) ∑_r=1^n x^4+5x^3+6r^2+r^3+5x^2+6x ∑_r=1^n x^4+6r^3+11r^2+6r 130(6n^5+15n^4+10n^3 n)+6 ...

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Byju's Answer

Standard X

Mathematics

Roots of Quadratic Equation

Find the sum ...

Question

Find the sum $\sum_{r = 1}^{n} r (r + 1) (r + 2) (r + 3)$

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Solution

$n \sum r = 1 r (r + 1) (r + 2) (r + 3)$
$n \sum r = 1 (r^{2} + r) (r^{2} + 5 r + 6)$
$n \sum r = 1 x^{4} + 5 x^{3} + 6 r^{2} + r^{3} + 5 x^{2} + 6 x$
$n \sum r = 1 x^{4} + 6 r^{3} + 11 r^{2} + 6 r$
$\frac{1}{30} (6 n^{5} + 15 n^{4} + 10 n^{3} - n) + 6 \frac{n^{2} (n + 1)^{2}}{4} + \frac{11 n (n + 1) (2 n + 1)}{6} + \frac{6 n (n + 1)}{2}$
$\frac{1}{30} (6 n^{5} + 15 n^{4} + 10 n^{3} - n) + \frac{3}{2} n^{2} (n^{2} + 1 + 2 n) + \frac{11 n}{6} (2 n^{2} + 3 n + 1) + 3 (n^{2} + n)$
$\frac{1}{30} (6 n^{5} + 15 n^{4} + 10 n^{3} - n) + \frac{3}{2} (n^{4} + 2 n^{3} + n^{2}) + \frac{11}{6} (2 n^{3} + 3 n^{2} + n) + 3 (n^{2} + n)$

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Find the sum ∑nr=1r(r+1)(r+2)(r+3)

n∑r=1r(r+1)(r+2)(r+3)n∑r=1(r2+r)(r2+5r+6)n∑r=1x4+5x3+6r2+r3+5x2+6xn∑r=1x4+6r3+11r2+6r130(6n5+15n4+10n3−n)+6n2(n+1)24+11n(n+1)(2n+1)6+6n(n+1)2130(6n5+15n4+10n3−n)+32n2(n2+1+2n)+11n6(2n2+3n+1)+3(n2+n)130(6n5+15n4+10n3−n)+32(n4+2n3+n2)+116(2n3+3n2+n)+3(n2+n)

Find the sum $\sum_{r = 1}^{n} r (r + 1) (r + 2) (r + 3)$