find T
8 T9 T10 for each of the following a) 5,8,11,14,... b) 3,5,7,9,... c) -1,2,5,8... d) 4,-1,-6,-11,... e)2,31\2 ,5,61\2
Solve the question
a) The given pattern is increasing by 3 each time. So, we can write the sequence as:
5, 8, 11, 14, ...
To find T9 and T10, we can use the formula for arithmetic sequence:
Tn = a + (n - 1) × d
For T9:
T9 = 5 + (9 - 1) × 3
T9 = 5 + 8 × 3
T9 = 5 + 24
T9 = 29
For T10:
T10 = 5 + (10 - 1) × 3
T10 = 5 + 9 × 3
T10 = 5 + 27
T10 = 32
So, T9 = 29 and T10 = 32.
b) The given pattern is increasing by 2 each time. So, we can write the sequence as:
3, 5, 7, 9, ...
To find T9 and T10:
T9 = 3 + (9 - 1) × 2
T9 = 3 + 8 × 2
T9 = 3 + 16
T9 = 19
T10 = 3 + (10 - 1) × 2
T10 = 3 + 9 × 2
T10 = 3 + 18
T10 = 21
So, T9 = 19 and T10 = 21.
c) The given pattern is increasing by 3 each time. So, we can write the sequence as:
-1, 2, 5, 8, ...
To find T9 and T10:
T9 = -1 + (9 - 1) × 3
T9 = -1 + 8 × 3
T9 = -1 + 24
T9 = 23
T10 = -1 + (10 - 1) × 3
T10 = -1 + 9 × 3
T10 = -1 + 27
T10 = 26
So, T9 = 23 and T10 = 26.
d) The given pattern is decreasing by 5 each time. So, we can write the sequence as:
4, -1, -6, -11, ...
To find T9 and T10:
T9 = 4 + (9 - 1) × -5
T9 = 4 + 8 × -5
T9 = 4 - 40
T9 = -36
T10 = 4 + (10 - 1) × -5
T10 = 4 + 9 × -5
T10 = 4 - 45
T10 = -41
So, T9 = -36 and T10 = -41.
e) The given pattern alternates between adding 1 and adding 2. So, we can write the sequence as:
2, 3.5, 5, 6.5, ...
To find T9 and T10:
T9 = 2 + (9 - 1) × 1.5
T9 = 2 + 8 × 1.5
T9 = 2 + 12
T9 = 14
T10 = 2 + (10 - 1) × 1.5
T10 = 2 + 9 × 1.5
T10 = 2 + 13.5
T10 = 15.5
So, T9 = 14 and T10 = 15.5.
To find the values of T for each of the given sequences, we need to determine the pattern or rule that governs each sequence.
a) Sequence: 5, 8, 11, 14, ...
Rule: The terms increase by 3.
To find the value of T, we need to determine the position of T in the sequence. Assuming the given sequence starts with the first term at position 1, we can see that T would be the 3rd term in this sequence.
Therefore, T = 5 + (3 * 2) = 5 + 6 = 11.
b) Sequence: 3, 5, 7, 9, ...
Rule: The terms increase by 2.
Similar to the previous case, T would be the 4th term in this sequence.
Therefore, T = 3 + (2 * 3) = 3 + 6 = 9.
c) Sequence: -1, 2, 5, 8, ...
Rule: The terms increase by 3.
Again, T would be the 4th term in this sequence.
Therefore, T = -1 + (3 * 3) = -1 + 9 = 8.
d) Sequence: 4, -1, -6, -11, ...
Rule: The terms decrease by 5.
This time, T would be the 2nd term in the sequence.
Therefore, T = 4 - (5 * 1) = 4 - 5 = -1.
e) Sequence: 2, 31/2, 5, 61/2, ...
Rule: The terms alternate between increasing by 31/2 and decreasing by 31/2.
Here, T would be the 4th term in this sequence.
Therefore, T = 2 + (31/2 * 3) = 2 + (31/2 * 3/1) = 2 + (93/2) = 2 + 46.5 = 48.5.
So to summarize:
a) T = 11
b) T = 9
c) T = 8
d) T = -1
e) T = 48.5
T9 = a+8d
T10 = a+9d
surely you can figure out a (the same as T1)
to find the common difference, just subtract any given term from the following term.