Mandy used the rule “Add 6” to make a pattern. She started with 20 and wrote the next 5 numbers in her pattern.
2 answers:
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Answer:
105th term of given series is
Step-by-step explanation:
Given series is
As we can see,
Also,
hence, we can say given series is in arithmetic progression,
with common difference,
As given in question the nth term in A.P is given by
since we have to find the 105th term, so we can write
Hence, the 105th term of given series of A.P is .
Answer:
Area = 13.15 square units
Step-by-step explanation:
Let the given vertices be represented as follows:
A(2, -1, 1) = 2i - j + k
B(5, 1, 4) = 5i + j + 4k
C(0, 1, 1) = 0i + j + k
D(3, 3, 4) = 3i + 3j + 4k
(i) Let's calculate the vectors of all the sides:
AB = B - A = (5i + j + 4k) - (2i - j + k)
AB = 5i + j + 4k - 2i + j - k [Collect like terms]
AB = 3i + 2j + 3k
BC = C - B = (0i + j + k) - (5i + j + 4k)
BC = 0i + j + k - 5i - j - 4k [Collect like terms]
BC = -5i + 0j - 3k
CD = D - C = (3i + 3j + 4k) - (0i + j + k)
CD = 3i + 3j + 4k - 0i - j - k [Collect like terms]
CD = 3i + 2j + 3k
DA = A - D = (2i - j + k) - (3i + 3j + 4k)
DA = 2i - j + k - 3i - 3j - 4k [Collect like terms]
DA = -i - 4j - 3k
AC = C - A = (0i + j + k) - (2i - j + k)
AC = 0i + j + k - 2i + j - k [Collect like terms]
AC = -2i + 2j
BD = D - B = (3i + 3j + 4k) - (5i + j + 4k)
BD = 3i + 3j + 4k - 5i - j - 4k [Collect like terms]
BD = -2i + 2j
(ii) From the results in (i) above, we can deduce that;
AB = CD This implies that AB || CD [AB is parallel to CD]
AC = BD This implies that AC || BD [AC is parallel to BD]
(iii) Therefore, ABDC is a parallelogram since opposite sides (AB and CD) are parallel. Hence, the points are vertices of a parallelogram
<u>Now let's calculate the area</u>
To find the area of the parallelogram, we find the magnitude of the cross product of any two adjacent sides.
In this case, we'll choose AB and AC
Area = |AB X AC|
Where;
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AB X AC = i(0 - 6) - j(0 + 6) + k(6 + 4)
AB X AC = - 6i - 6j + 10k
|AB X AC| =
|AB X AC| =
|AB X AC| = 13.15
Area = 13.15 square units.
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<u>PS: </u> ACBD is also a parallelogram. The diagram has also been attached to this response.
