The first and last term of AP are 79and -5 if the sun is 814 find
The number of terms on the AP
The common difference
sum(n) = n(first + last)/2
814 = n(79-5)/2
814 = 37n
n = 22
term(22) = a + 21d
-5 = 79 + 21d
d = -4
To find the number of terms in an arithmetic progression (AP), we can use the formula:
Number of terms (n) = (last term - first term) / common difference + 1
Given that the first term (a₁) is 79, the last term (aₙ) is -5, and the sum of the AP is 814, we need to find the common difference (d) first.
Sum of an AP (S) = (n/2) * (2a₁ + (n-1)d)
Substituting the values, we have:
814 = (n/2) * (2*79 + (n-1)d)
814 = (n/2) * (158 + (n-1)d)
To calculate the common difference, we can rearrange the equation to solve for (n-1)d:
814 = (n/2) * (158 + (n-1)d)
814 * 2 = n * (158 + (n-1)d)
1628 = 158n + n² - nd
We know that n = (aₙ - a₁) / d + 1, so we can rewrite the equation as:
1628 = 158 * [(aₙ - a₁)/d + 1] + [(aₙ - a₁)/d + 1]² - [(aₙ - a₁) / d] * d
Now we can substitute the given values:
1628 = 158 * [(−5 − 79)/d + 1] + [(−5 − 79)/d + 1]² - [(−5 − 79)/d] * d
Simplifying the equation:
1628 = 158 * (-84/d + 1) + (-84/d + 1)² + 84/d * d
Expanding the equation:
1628 = -13272/d² + 166 * d - 13904/d + 676 + 2(-84/d + 1)(84/d * d)
Simplifying further:
1628 = -13272/d² + 166d - 13904/d + 676 - 168 * d /d + 168/d²
Combining like terms:
1628 = -13272/d² - 2d - 13904/d + 676 + 168/d²
Rearranging the equation:
0 = -13272/d² - 2d - 13904/d + 676 + 168/d² - 1628
Combining like terms:
0 = -126596/d² - 2d + 168/d² - 952
Solving the equation:
0 = -126596 - 2d(d²) + 168 - 952(d²)
0 = -126596 - 2d³ + 168d² - 952
Since this equation has a cubic term and does not have a simple solution, we will use numerical methods such as Newton's method or a graphing calculator to find the value of d.
Once we have the common difference (d), we can substitute it back into the formula for the number of terms to find 'n'.
To find the number of terms in the arithmetic progression (AP) and the common difference, we can use the formula:
nth term = first term + (n-1) * common difference
In this case, we have:
First term (a1) = 79
Last term (an) = -5
Sum of terms (S) = 814
To find the number of terms (n), we can substitute the values into the above formula and solve for n.
-5 = 79 + (n - 1) * common difference ------(1)
To find the common difference, we can use the formula for the sum of an arithmetic series:
S = (n/2) * (first term + last term)
Substituting the given values, we have:
814 = (n/2) * (79 + (-5)) ------(2)
Now, we can solve these two equations simultaneously to find the values of n and the common difference.
Let's solve equation (1) for common difference:
-5 - 79 = (n - 1) * common difference
-84 = (n - 1) * common difference
Now, let's substitute this value into equation (2):
814 = (n/2) * (79 - 84)
814 = (n/2) * (-5)
-1628 = n * -5
n = -1628 / -5
n = 325.6
Since the number of terms must be a whole number, we round it up to the nearest integer:
n = 326
Now, let's substitute this value of n back into equation (1) to find the common difference:
-5 = 79 + (326 - 1) * common difference
-5 = 79 + 325 * common difference
-5 - 79 = 325 * common difference
-84 = 325 * common difference
common difference = -84 / 325
common difference ≈ -0.258
Therefore, the number of terms in the AP is 326 and the common difference is approximately -0.258.