Given the first and last terms in the sequence , calculate the sum using
= (a + l)
where a is the first term and l the last term
Here a = 17 and l = 35
To calculate the number of terms use the n th term formula
= a + (n - 1)d
where d is the common difference
a = 17 and d = 19 - 17 = 2 and = 35, thus
17 + 2(n - 1) = 35 ( subtract 17 from both sides )
2(n - 1) = 18 ( divide both sides by 2 )
n - 1 = 9 ( add 1 to both sides )
n = 10
= 5(17 + 35) = 5 × 52 = 260
common difference (d ) = 19-17 = 2
a+(n-1)d = 35
17+(n-1)2 = 35
(n-1)2 = 35-17
n-1 = 18/2
n = 9+1 = 10
Sum of the arithmetic sequence
= 10/2 [ 2(17)+ (10-1)2]
= 5× 52
When there is no decomposition, the plant is deficient in minerals.
x represent the number of trips that the larger truck makes
so, we are given
If the smaller truck makes 18 more trips than the larger
number of trips for larger truck =x
number of trips for smaller truck =x+18
The capacities of two trucks are 3 tons and 4 tons respectively
so, capacity of smaller truck per trip =3 tons / trip
total ton carried by smaller truck = (number of trips for smaller truck)*(capacity of smaller truck per trip)
now, we can plug values
total ton carried by smaller truck =3*(x+18)
total ton carried by larger truck = (number of trips for larger truck)*(capacity of larger truck per trip)
total ton carried by larger truck =4x
it can deliver 12 more tons of freight than the larger
so, we get
we can move 4x on left side
9514 1404 393
n = 8
"By inspection" is an appropriate method.
We are asked to compare the expressions
and find the value(s) of n that makes them equal. <em>By inspection</em>, we see that n=8 will make these expressions equal. We also know that both expressions will be zero when n=0.
More formally, we could write ...
n^2 = 8n . . . . the two formulas give the same value
n^2 -8n = 0 . . . . rearrange to standard form
n(n -8) = 0 . . . . . factor
Using the zero product rule, we know the solutions will be the values of n that make the factors zero. Those values are ...
n = 0 . . . . . makes the factor n = 0
n = 8 . . . . . makes the factor (n-8) = 0
Generally, we're not interested in "trivial" solutions (n=0), so the only value of n that is of interest is n = 8.
A lot of times, I find a graphing calculator to be a quick and easy way to find function argument values that make expressions equal.