the first yerm of an A.p is 7 and 10th term is twice the second term calculate the 19th term
B. The sum of the first 28th terms
To find the common difference in the arithmetic progression (A.P), we can use the formula:
Tn = a + (n-1)d
Where Tn is the nth term, a is the first term, n is the term number, and d is the common difference.
Given that the first term (a) is 7 and the 10th term is twice the second term:
T10 = 7 + (10-1)d = 7 + 9d
T10 = 2(T2) = 2(a + (2-1)d) = 2(a + d) = 2(7 + d) = 14 + 2d
Now, we can solve the equation:
7 + 9d = 14 + 2d
Subtracting 2d from both sides:
7 + 7d = 14
Subtracting 7 from both sides:
7d = 7
Dividing both sides by 7:
d = 1
Therefore, the common difference is 1.
To find the 19th term, we can substitute the values into the formula:
T19 = a + (19-1)d
T19 = 7 + (18)(1)
T19 = 7 + 18
T19 = 25
Therefore, the 19th term is 25.
To calculate the sum of the first 28 terms, we can use the sum formula for an A.P:
Sn = (n/2)(2a + (n-1)d)
Given that n = 28, a = 7, and d = 1, we can substitute these values into the formula:
S28 = (28/2)(2(7) + (28-1)(1))
S28 = 14(14 + 27)
S28 = 14(41)
S28 = 574
Therefore, the sum of the first 28 terms is 574.
To find the 19th term of the arithmetic progression (AP), we need to determine the common difference (d) first.
Let's use the formula for the nth term of an AP:
an = a1 + (n - 1)d
Given information:
a1 = 7 (first term)
a10 = 2a2 (tenth term is twice the second term)
To find the common difference (d), we can use the formula:
a10 = a1 + (n - 1)d
2a2 = a1 + 9d
Since a10 is twice a2, we can replace a10 with 2a2:
2a2 = a1 + 9d
Now, we substitute the value of a1 from the given information:
2a2 = 7 + 9d
Next, we can solve for the common difference (d):
2a2 - 7 = 9d
9d = 2a2 - 7
d = (2a2 - 7)/9
Now, we can find the 19th term (a19) using the formula:
a19 = a1 + (n - 1)d
Substituting the values:
a19 = 7 + (19 - 1) * [(2a2 - 7)/9]
Simplifying further:
a19 = 7 + 18(2a2 - 7)/9
a19 = 7 + 2(2a2 - 7)
a19 = 7 + 4a2 - 14
a19 = 4a2 - 7
To find the sum of the first 28 terms, we can use the formula for the sum of an AP:
Sn = (n/2) * (a1 + an)
Substituting the given values:
S28 = (28/2) * (a1 + a28)
S28 = 14 * (7 + a28)
We need to find a28 to calculate the sum of the first 28 terms.
Using the same formula for the nth term of an AP:
a28 = a1 + (28 - 1) * d
a28 = 7 + 27d
Substituting the value of d we found earlier:
a28 = 7 + 27[(2a2 - 7)/9]
Simplifying further:
a28 = 7 + 6(2a2 - 7)
a28 = 7 + 12a2 - 42
a28 = 12a2 - 35
Substituting this value into the sum formula:
S28 = 14 * (7 + 12a2 - 35)
S28 = 14 * (12a2 - 28)
S28 = 168a2 - 392
Now, we have calculated the 19th term (a19) and the sum of the first 28 terms (S28) of the arithmetic progression.
To find the 19th term of an arithmetic progression (A.P.), we need to determine the common difference between the terms.
Given:
First term (a) = 7
10th term (T10) = 2 * (a + d) (where d is the common difference)
We can use this information to express the 10th term in terms of the first term and the common difference.
T10 = a + 9d (since the 10th term is a + 9d)
2(a + d) = a + 9d (from the given information)
2a + 2d = a + 9d
a = 7d
Now, we know that the first term (a) is equal to 7d.
To find the common difference (d), we substitute the value of the first term (7d) into the expression for the 10th term.
7d = 7 + 9d
-2d = 7
d = -7/2
Now that we have the common difference (d), we can find the 19th term of the A.P.
T19 = a + 18d (since the 19th term is a + 18d)
T19 = 7d + 18d
T19 = 25d
Plugging in the value of d:
T19 = 25 * (-7/2)
T19 = -25/2
Therefore, the 19th term of the A.P is -25/2.
To calculate the sum of the first 28 terms, we can use the formula for the sum of an A.P:
Sum of n terms (Sn) = (n/2) * (2a + (n - 1)d)
Given:
n = 28
a = 7
d = -7/2
Plugging in these values into the formula:
S28 = (28/2) * (2(7) + (28 - 1)(-7/2))
S28 = 14 * (14 - 27)
S28 = 14 * (-13)
S28 = -182
Therefore, the sum of the first 28 terms of the A.P. is -182.