The Fibonacci sequence is defined by 1 = a₁ = a₂ and a_n = a_{n - 1} + a_n = a_{n - 2}  n > 2. Find \(\frac { { a }_{ n+1 } }{ { a }_{ n } }\), for n = 1, 2, 3, 4 , 5.

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(a)

Write the first five terms of each of the following sequence whose nth terms are a_n = \(\frac { n }{ n+1 } \)

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(a)

Find the indicated terms in each of the sequence, where nth terms are given.
a_n = 5n - 3, a₁₂, a₁₅

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(a)

Find the indicated terms in each of the sequence, where nth terms are given.
(i) b_n = \({ (-1) }^{ n }({ n }^{ 2 }-1)\), b_7, b₁₃ 
(ii) \(a_{n}=\frac{n^{2}}{2^{n}}, a_{7}\)

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(a)

Write the first five terms of each of the sequence and obtain the corresponding series. a₁ = 3, a_n = 3a_n-1+2, for all n >1

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(a)

Write the first five terms of each of the sequence and obtain the corresponding series. a₁ = a₂ = 2, a_n = a_n-1 -1, n > 2

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(a)

Show that the sequence a_n  defined by a_n = 2n² +1, is not an AP.

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(a)

The nth term of a sequence is 3n - 2. Is the sequence an AP ? If it is an AP, then find its 6th term.

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(a)

Find the sum of 20 terms of an AP, whose first term is 3 and last term is 57.

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(a)

The (m+n)^th and (m -n)th terms of G.P are P and q respectively. Show that the m^th and n^th terms are \(\sqrt { pq } \) and \(p\left( \frac { p }{ q } \right) ^{ \frac { m }{ 2n } }\) respectively

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(a)

Determine the number of terms in the A.P. 4, 8, 12, 288. Also find its 18th term from the end.

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(a)

Which term of the C.P. 2, 8, 32, ...... is 32768 ?

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(a)

The fourth term of a C.P. is 4. Find the product of its first seven terms.

3

(a)

Find the sum to infinity of the series (x + y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + .......

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(a)

If S1, S2, S3, .... SP denote the sums of infinite geometric series whose first terms are 1, 2, 3, .... P respectively and whose common rations are \(\frac{1}{2},\frac{1}{3},\frac{1}{4},....,\frac{1}{P-1}\) respectively Prove that \(S_1+S_2+S_3+..+S_P=\frac{P(P+3)}{2}\)

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(a)