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Arithmetic Sequence?

1. Explain ow to write an explicit formula for the general term of the sequence [-4,2,8,14,...]

It is an A.P. with a(1) = - 4 and d= 6.

a(n) = a(1) + (n-1)d = - 4 + (n-1)6= 6n - 10 Answer

Verify by putting n= 1,2,3 etc OK

2. Explain why (n-1)d instead of nd is used to find the nth term of an arithmetic sequence.

a(1) = a(1) + 0 d, a(2) = a(1) + 1 d, a(3) = a(1) + 2 d,

a(4) = a(1) + 3 d, etc Observe the pattern and that leads to the answer of your question.

3. Describle how the arithmetic mean of 4 and 20 and the three arithmetic means between 4 and 20 are related

Arithmetic mean A between a and b is given by A=(1/2)(a+b)

In our case A = (1/2)(4+20) =12.

By definition a, A, b are in A.P.

Let the three means between c and d be E, F and G,

Then by definition c, E, F, G, d are in A.P.

In our case 4, E, F, G, 20 are in A.P. So 20 is the fifth term of A.P.

So: 20 = 4 + 4d Or d= 4. So: E=8, F= 12, G = 16 (each of these terms is obtained by by adding the common difference d to its previous term.

Let the original sequence three numbers are a, a + 3, and a + 6. If the first number is increased by 1, the second increased by 6 , and the third by 19 So, the geometric sequence three numbers are a + 1, a + 9, and a + 25. Above numbers following geometric sequence so, t₂ / t₁ = t₃ / t₂ (a + 9)/(a + 1) = (a + 25)/(a + 9) Cross multiplication. (a + 9)(a + 9) = (a + 25)(a + 1) FOIL method: the product of two binomials is the sum of the products of the First terms, the Outer terms, the Inner terms and the Last terms. a2 + 9a + 9a + 81 = a2 + 1a + 25a + 25 a2 + 18a + 81 = a2 + 26a + 25 Subtract a2 from each side. 18a + 81 = 26a + 25. Subtract 18a from each side. 81 = 8a + 25 Subtract 25 from each side. 8a = 56 Divide each side by 8. a = 7. Therefore the original sequence is 7, 10, 13. The geometric sequence is 8, 16, 32.