Given the AP 1,5,..........69 find the common difference
The number of terms in the sequence
d = 5-1 = 4
n = (69-1)/4 + 1 = 18
To find the common difference of an arithmetic progression (AP), we need to find the difference between any two consecutive terms.
In this case, we are given the AP 1, 5, ..., 69. Let's consider the first two terms, 1 and 5. The common difference is obtained by subtracting the first term from the second term:
5 - 1 = 4
Therefore, the common difference of the given AP is 4.
To find the number of terms in the sequence, we can use the formula for the nth term of an AP:
an = a + (n-1)d
where 'an' is the nth term, 'a' is the first term, 'n' is the number of terms, and 'd' is the common difference.
In this case, we are given that the first term is 1, and the common difference is 4. We need to find the value of 'n'. The last term of the sequence is 69.
69 = 1 + (n-1)4
Simplifying the equation:
69 = 1 + 4n - 4
Combining like terms:
69 = 4n - 3
Adding 3 to both sides:
72 = 4n
Dividing both sides by 4:
n = 18
Therefore, the number of terms in the given AP is 18.
To find the common difference in an arithmetic progression (AP), subtract the first term from the second term.
Common difference = second term - first term = 5 - 1 = 4.
To find the number of terms in the sequence, we can use the formula for the nth term of an AP:
nth term = first term + (n - 1) × common difference,
where nth term represents the value of the term, n represents the position of the term in the sequence (starting from 1), first term represents the initial term, and common difference represents the common difference.
We can use this formula to find the value of n when the nth term is 69:
69 = 1 + (n - 1) × 4.
Simplifying the equation:
68 = 4n - 4.
Adding 4 to both sides:
72 = 4n.
Dividing both sides by 4:
n = 18.
Therefore, the number of terms in the sequence is 18.