Prove Propositions 32 and 33 of the Maasei Hoshev: a) 1 + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + ... + n) =    1 2 + 32 + ... + n 2 , if n is odd; 2 2 + 42 + ... + n 2 , if n is even.

What is the value of x in the equation 6(x + 1) – 5x = 8 + 2(x – 1)?

Write two subtraction equations that are equivalent to 5 + 12 = 17.

The average human heart beats 1.15 \cdot 10^51.15⋅10 5 1, point, 15, dot, 10, start superscript, 5, end superscript times per day. there are 3.65 \cdot 10^23.65⋅10 2 3, point, 65, dot, 10, start superscript, 2, end superscript days in one year.how many times does the heart beat in one year? write your answer in scientific notation, and round to one decimal place.