Given the A.P -6, 2½,1...... 71 find the
a. Common difference
b. Numbers of terms
To find the common difference of an arithmetic progression (A.P), we need to subtract any two consecutive terms. In this case, we can subtract the second term (2½) from the first term (-6):
Common difference = 2½ - (-6) = 2½ + 6 = 8½
Therefore, the common difference is 8½.
To find the number of terms in an arithmetic progression, we can use the formula:
n = (last term - first term) / common difference + 1
In this case, the last term is given as 71, the first term is -6, and the common difference we found is 8½.
n = (71 - (-6)) / 8½ + 1
n = 77 / 8½ + 1
n = 77 / 8.5 + 1
n ≈ 9 + 1
n = 10
Therefore, there are 10 terms in the arithmetic progression.
To find the common difference and number of terms in the arithmetic progression (A.P) -6, 2½, 1, ...... 71, we need to first examine the pattern of the progression.
a. Common difference:
In an arithmetic progression, the common difference (d) is the constant difference between any two consecutive terms.
Let's calculate the difference between consecutive terms:
2½ - (-6) = 2½ + 6 = 8½
1 - 2½ = -1½
71 - 1 = 70
From the differences calculated, we can see that the common difference appears to be varying. It is 8½ between the first two terms, then -1½ between the second and third terms, and finally 70 between the third and fourth terms.
So, there is no constant common difference in the given arithmetic progression.
b. Number of terms:
Given the A.P. -6, 2½, 1, ..., 71, we can count the terms to determine the number of terms.
Counting the terms from -6 to 71, we find:
-6, 2½, 1, ..., 71
There appears to be a total of 31 terms in this arithmetic progression.
Therefore:
a. The common difference is not constant.
b. The number of terms is 31.
To solve the problem, we need to find the common difference and the number of terms in the given arithmetic progression (AP).
a. Common difference:
The common difference (d) in an AP refers to the constant difference between any two successive terms. To find the common difference, we can subtract any term from its preceding term.
In the given AP, the first term (a) is -6, and the second term (b) is 2½.
To find d:
d = b - a
d = 2½ - (-6)
d = 2½ + 6 (change the subtraction to addition by changing the sign of the negative value)
d = 8½
Therefore, the common difference is 8½.
b. Number of terms:
To find the number of terms in an AP, we need to determine the position of the last term, which in this case is 71.
To find the position of the last term (n), we can use the formula for the nth term of an AP:
an = a + (n - 1) * d
In this case, we know that the last term (an) is 71. Plugging the values into the formula, we have:
71 = -6 + (n - 1) * 8½
Simplifying the equation:
71 = -6 + 8½n - 8½
Multiply 8½ by n and -8½:
71 = 8½n - 14
Add 14 to both sides of the equation:
71 + 14 = 8½n
85 = 8½n
Now, let's divide both sides of the equation by 8½ to solve for n:
85 ÷ 8½ = n
Dividing 85 by 8½:
n = 10
Therefore, the number of terms in the AP is 10.