Expand and simplify the following:
4
∑
i=1(i+3/i+4)
(1+3/1+4)+(2+3/2+4)+(3+3/3+4)+(4+3/4+4)
4/5+5/6+6/7+7/8
I am stuck here please help me.
Not certain what your issue is, in your google search window,
4/5+5/6+6/7+7/8=
3.36547619 is given as an approximate answer.
It gets a bit messy, but since the LCD is 5*6*7*4=840
4/5+5/6+6/7+7/8
= (4*168 + 5*140 + 6*120 + 7*105)/840 = 2827/840
To expand and simplify the given expression, we need to evaluate the summation by plugging in the values of i and performing the calculations.
The given expression is:
∑(i=1 to 4) (i+3)/(i+4) = (1+3)/(1+4) + (2+3)/(2+4) + (3+3)/(3+4) + (4+3)/(4+4)
Let's calculate each term step by step:
Term 1: (1+3)/(1+4) = 4/5
Term 2: (2+3)/(2+4) = 5/6
Term 3: (3+3)/(3+4) = 6/7
Term 4: (4+3)/(4+4) = 7/8
Now, sum up all the terms:
Sum = 4/5 + 5/6 + 6/7 + 7/8
To simplify the sum, we need to find a common denominator for all the fractions. In this case, the common denominator is 168. Let's rewrite each fraction with this denominator:
Sum = (4/5)(34/34) + (5/6)(28/28) + (6/7)(24/24) + (7/8)(21/21)
= (136/170) + (140/168) + (144/168) + (147/168)
Next, let's combine the fractions:
Sum = (136 + 140 + 144 + 147)/168
= 567/168
Lastly, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 7:
Sum = (81/24)
Therefore, the expanded and simplified expression is:
∑(i=1 to 4) (i+3)/(i+4) = 81/24 or simplified as 27/8.