Search results
Results from the WOW.Com Content Network
Write down the initial sequence: color(red)(1), 3, 6, 10, 15 Write down the sequence of differences ...
These are the triangular numbers - each term in the sequence being the sum of the first #n# positive integers:. #T_1 = 1 = 1#
A finite sequence does not determine a unique formula, but in this particular case there are enough terms to see an intended pattern. We can derive an explicit formula by examining sequences of differences. The given sequence is: #color(blue)(0), 1, 3, 6, 10# Write down the sequence of differences of that sequence: #color(blue)(1), 2, 3, 4#
#a_1# is the first element of a arithmetic sequence, #a_2# will be by definition #a_2=a_1+d#, #a_3=a_2+d#, and so on. Example1: 2,4,6,8,10,12,....is an arithmetic sequence because there is a constant difference between two consecutive elements (in this case 2) Example 2:
Starting with 1 you can begin to see a pattern develop.... #a_1=1# #a_2=a_1+2=1+2=3# #a_3=a_2+3=3+3=6# #a_4=a_3+4=6+4=10# #a_5=a_4+5=10+5=15# #vdots# #a_n=a_(n-1)+n# Since every term is merely the sum of the previous term plus the next counting number we have. #a_n=sum_(k=1)^nk=(n(n+1))/2# To test this we use the first formula for #a_(n+1)#
55 Add the first and the last numbers of the sequence and repeat for the other sets of numbers. 1 + 10 = 11 2 + 9 = 11 3 + 8 = 11 4 +7 = 11 5 + 6 = 11 This gives 5 sets of numbers that equal 11. 5 xx 11 = 55
32.4 Consider the ratios of consecutive numbers in the sequence as follows: 1.2/0.4 = 3 3.6/1.2=3 10.8/3.6=3 Hence, the sequence is a geometric progression with a common ratio of 3. Therefore the next number in the sequence would be: 10.8 xx 3= 32.4
T_n = 2n -1 This is clearly an arithmetic sequence because the terms differ by 2 each time. To find the nth term rule we need: a value for the first term , a and a value for the common difference d . These values are then plugged into the formula: T_n = a + (n-1)d a = 1 and d = 2 T_n =a + (n-1)d T_n = 1 + (n-1)2 T_n = 1 + 2n -2 T_n = 2n -1
3, 4, 6, 10, 18 When a sequence is defined recursively, the previous term is used to find the next term. Start by using a_1 to find a_2 (second term). Substitute a_1 for a_k. a_(1+1)=2(a_1-1) a_2=2(a_1-1) Because a_1=3, substitute a_1 for 3 to find a_2. a_2=2(3-1) a_2=2(2) a_2=4 This is your second term! Now use a_2 to find a_3 just like how you used a_1 to find a_2. a_(2+1)=2(a_2-1) a_3=2(4-1 ...
21 Look at the difference between two,consecutive numbers: the first two differ by three, the second and the third by four, the third and the fourth by five. So, the next number will be six more than the last one, i.e. 15+6=21