The third term of an A.P is 8 and 9th term of the A.P.exceeds three times the third term by 2.find the sum of its first 19th terms.
a= 2
S19=551
how in first 26 came, sir pls..can u explain me....
sir pls can u tell me in first how 26 came...?
how 26 came ??
To find the sum of the first 19 terms of an arithmetic progression (A.P.), we need to know the first term (a), the common difference (d), and the number of terms (n).
Given:
The third term (a3) of the A.P. is 8.
The ninth term (a9) exceeds three times the third term by 2.
We are asked to find the sum of the first 19 terms (S19).
Step 1: Finding the common difference (d)
We know that the difference between any two consecutive terms in an A.P. is constant. To find the common difference (d), we can use the formula:
d = a3 - a2
Since the third term (a3) is given as 8, and the second term (a2) can be found using the formula:
a2 = a3 - d,
we can substitute the given value of a3 into the formula to find d.
Step 2: Finding the ninth term (a9)
We are told that the ninth term (a9) exceeds three times the third term by 2. Using the formulas, we can express this relationship as:
a9 = 3a3 + 2
substituting the given value of a3, we have:
a9 = 3 * 8 + 2
Step 3: Finding the first term (a)
To find the first term (a) of the A.P., we can use the formula:
a = a3 - 2d
Step 4: Finding the common difference (d)
Substituting the value of a2 (which is a3 - d) and a3 (which is given as 8) into the equation a9 = 3a3 + 2, we can solve for d:
a9 = 3 * 8 + 2
a9 = 24 + 2
a9 = 26
Step 5: Finding the sum of the first 19 terms (S19)
To find the sum of the first 19 terms (S19) of the A.P., we can use the formula:
S19 = (n/2)(2a + (n-1)d)
Substituting the values of a, n, and d into the equation, we can calculate S19.
a+2d = 8
a+8d = 3(a+2d)+2 = 26
so, 6d = 18, and d=3
Now you can find a, and answer the question.
a+2d=8
a+8d=3(a+2d)+2=26
6d=18
d=6
put d in a+2d= 8
a+2(3)=8
a+6=8
a=8-2
a=2
Sn= n/2[2a+(n-1)d]
=n/2[2(2)+(n-1)3]
n/2[4+3n-3]
2n+3n-3
5n+3
n=5/3