The 43rd term of an Ap is 26. Find the first term of the progression, given that it's common difference is 1/2 . Find the number of terms is an Ap given that it's last term
To find the first term of the arithmetic progression (AP), we can use the formula:
an = a1 + (n-1)d
Where an is the nth term, a1 is the first term, n is the number of terms, and d is the common difference.
We are given that the 43rd term (an) is 26, and the common difference (d) is 1/2.
Substituting these values into the formula, we get:
26 = a1 + (43-1)(1/2)
26 = a1 + 42/2
26 = a1 + 21
Subtracting 21 from both sides, we get:
a1 = 26 - 21
a1 = 5
Therefore, the first term of the progression is 5.
To find the number of terms in the AP, we can use the formula:
an = a1 + (n-1)d
Again, we are given that the last term (an) is known, and the common difference (d) is known. However, the first term (a1) is not given.
Since we know that the 43rd term is 26, and the common difference is 1/2, we can substitute these values into the formula to find the value of a1:
26 = a1 + (43-1)(1/2)
26 = a1 + 42/2
26 = a1 + 21
Subtracting 21 from both sides, we get:
a1 = 26 - 21
a1 = 5
We now know that the first term (a1) is 5. However, we still need to find the value of n, the number of terms.
To find n, we can rearrange the formula:
an = a1 + (n-1)d
To isolate n, we can subtract a1 from both sides and divide both sides by d:
an - a1 = (n-1)d
n-1 = (an - a1)/d
n = ((an - a1)/d) + 1
Plugging in the known values, we get:
n = ((26 - 5)/(1/2)) + 1
n = (21/(1/2)) + 1
n = 21/(1/2) + 1
n = 21 * 2 + 1
n = 42 + 1
n = 43
Therefore, the number of terms in the arithmetic progression is 43.