Find n,if 242 is the nth term of 2,4,10,14.....
In the given sequence, the difference between consecutive terms is not constant. To find a pattern, let's look at the differences between the terms:
1st difference:
4 - 2 = 2
2nd difference:
10 - 4 = 6
3rd difference:
14 - 10 = 4
It appears that the differences between the differences are constant. Therefore, it suggests that the sequence has a quadratic pattern.
Let's use the formula for the nth term of a quadratic sequence:
nth term = an^2 + bn + c
To find the values of a, b, and c, we can substitute the first three terms of the sequence into the formula:
2 = a(1)^2 + b(1) + c
4 = a(2)^2 + b(2) + c
10 = a(3)^2 + b(3) + c
Simplifying these equations, we get:
a + b + c = 2
4a + 2b + c = 4
9a + 3b + c = 10
Solving this system of equations, we find that a = 2, b = -2, and c = 2.
Therefore, the formula for the nth term is:
nth term = 2n^2 - 2n + 2
Now, let's substitute 242 into this formula and solve for n:
242 = 2n^2 - 2n + 2
Simplifying this equation, we get:
2n^2 - 2n - 240 = 0
Dividing through by 2, we have:
n^2 - n - 120 = 0
Factoring this quadratic equation, we get:
(n - 12)(n + 10) = 0
Thus, n = 12 or n = -10.
Since n represents the position of the term in the sequence, it cannot be negative. Therefore, the value of n is 12.
To find the value of n in the sequence 2, 4, 10, 14..., we need to determine the pattern or rule first.
Looking at the sequence, we can see that each term is obtained by adding a specific value to the previous term. Specifically, the differences between the terms are 2, 6, 4,...
Now, let's calculate the differences between the terms:
4 - 2 = 2
10 - 4 = 6
14 - 10 = 4
From this, we can see that the differences alternate between 2 and 4. If we continue this pattern, the next difference should be 2 and the difference after that should be 6.
To find the nth term, we can use the formula:
nth term = first term + (n - 1) * common difference
Let's apply the formula to the given sequence:
2 + (n - 1) * 2 = 242
Simplifying the equation:
2n - 2 = 242 - 2
2n - 2 = 240
2n = 240 + 2
2n = 242
n = 242 / 2
n = 121
Therefore, 242 is the 121st term in the sequence.
To find the value of n for which 242 is the nth term, we first need to observe the pattern in the given sequence: 2, 4, 10, 14, ...
Looking at the terms, we can see that the pattern alternates between two different operations. Let's break it down:
Starting with 2, we add 2 to get the next term: 2 + 2 = 4.
Next, we add 6 to get the term after that: 4 + 6 = 10.
Now, there is a change in the pattern. Instead of adding 2, we add 4 to the previous term: 10 + 4 = 14.
We can continue this pattern to find the value of the nth term. Let's look at more terms:
10 + 6 = 16
16 + 2 = 18
18 + 6 = 24
24 + 4 = 28
28 + 6 = 34
34 + 2 = 36
36 + 6 = 42
42 + 4 = 46
46 + 6 = 52
52 + 2 = 54
54 + 6 = 60
60 + 4 = 64
From this pattern, we can see that every even-odd pair of numbers is increasing by 6, while every odd-even pair increases by 4.
Now, we can find n by starting from 2 and adding the appropriate amount each time until we reach 242.
2 + 2 = 4
4 + 6 = 10
10 + 4 = 14
14 + 6 = 20
20 + 6 = 26
26 + 4 = 30
30 + 6 = 36
36 + 4 = 40
40 + 6 = 46
46 + 4 = 50
50 + 6 = 56
56 + 4 = 60
60 + 6 = 66
66 + 4 = 70
70 + 6 = 76
76 + 4 = 80
80 + 6 = 86
86 + 4 = 90
90 + 6 = 96
96 + 4 = 100
100 + 6 = 106
106 + 4 = 110
110 + 6 = 116
116 + 4 = 120
120 + 6 = 126
126 + 4 = 130
130 + 6 = 136
136 + 4 = 140
140 + 6 = 146
146 + 4 = 150
150 + 6 = 156
156 + 4 = 160
160 + 6 = 166
166 + 4 = 170
170 + 6 = 176
176 + 4 = 180
180 + 6 = 186
186 + 4 = 190
190 + 6 = 196
196 + 4 = 200
200 + 6 = 206
206 + 4 = 210
210 + 6 = 216
216 + 4 = 220
220 + 6 = 226
226 + 4 = 230
230 + 6 = 236
236 + 4 = 240
240 + 6 = 246
Here, we can see that 242 is exceeded after the 48th term, so the value of n is 48.
Therefore, n = 48.