Find out a formula for the Nth term of each of the following sequences.
(i)4 7 10 13 16
(ii)1 3 5 7 9
(iii)20 17 14 11 8
(iv)1 4 9 16 25
(v)2 8 18 32 50
Find the sum of each of the following
16-20 4+11+18+25......:s 10
(i)4 7 10 13 16 N = 4 + (n-2)d
(ii)1 3 5 7 9 N = 2 + (n-1)d
(iii)20 17 14 11 8 N = 20 - (n-1)d
(iv)1 4 9 16 25 N = n^2
(v)2 8 18 32 50 N = 2n^2
i) term(n) = 3n +1
ii) term(n) = 2n - 1
iii) term(n) = -3n + 23
iv) term(n) = n^2
v) term(n) = 2(n^2)
To find the formula for the nth term of a sequence, we need to observe the pattern in the given sequence and identify the common difference or common ratio.
(i) The given sequence is 4, 7, 10, 13, 16. The common difference between consecutive terms is 3. So, the formula for the nth term of this sequence is given by:
a[n] = a[1] + (n - 1)d
where a[1] is the first term and d is the common difference.
In this case, a[1] = 4 and d = 3. Therefore, the formula for the nth term is:
a[n] = 4 + (n - 1)3 = 3n + 1
(ii) The given sequence is 1, 3, 5, 7, 9. The common difference between consecutive terms is 2. So, the formula for the nth term of this sequence is:
a[n] = a[1] + (n - 1)d
In this case, a[1] = 1 and d = 2. Therefore, the formula for the nth term is:
a[n] = 1 + (n - 1)2 = 2n - 1
(iii) The given sequence is 20, 17, 14, 11, 8. The common difference between consecutive terms is -3. So, the formula for the nth term of this sequence is:
a[n] = a[1] + (n - 1)d
In this case, a[1] = 20 and d = -3. Therefore, the formula for the nth term is:
a[n] = 20 + (n - 1)(-3) = -3n + 23
(iv) The given sequence is 1, 4, 9, 16, 25. The difference between consecutive terms is not constant. However, we can observe that the terms are perfect squares. So, the formula for the nth term of this sequence is:
a[n] = n^2
(v) The given sequence is 2, 8, 18, 32, 50. The difference between consecutive terms is not constant. However, we can observe that the difference between each term and its predecessor is increasing by 4 each time. So, the formula for the nth term of this sequence is:
a[n] = a[1] + (n - 1)(a[2] - a[1]) + (n - 1)(n - 2)d/2
In this case, a[1] = 2, a[2] = 8, and d = 4. Therefore, the formula for the nth term is:
a[n] = 2 + (n - 1)(8 - 2) + (n - 1)(n - 2)4/2 = 2 + 6(n - 1) + 2(n - 1)(n - 2)