$(2^{51}+2^{52}+2^{53}+2^{54}+2^{55})$ is divisible by

Q. Observe the following pattern for obtaining the sums, then find the sum of numbers from 781 to 790?
$1+2+3+4+5+6+7+8+9+10=55$
$11+12+13+14+15+16+17+18+19+20=155$
$21+22+23+24+25+26+27+28+29+30=255$
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$101+102+103+104+105+106+107+108+109+110=1055$
$251+252+253+254+255+256+257+258+259+260=2555$

Q. Three source of sound of equal intensities have frequencies 251, 252 and 253 Hz respectively. When sounded together, the number of beats heard per second will be

Q. Without actual division find ${27}^3-{25}^3-2^3$

Q. The middle term in the expansion of ${\left(\frac{2x^2}{3}+\frac{3}{2x^2}\right)}^{10}$ is
(a) 251
(b) 252
(c) 250
(d) none of these

Q. Simplify the expression: $\frac{{25}^4\times 5^2}{{25}^5}$ .