Answer:
And by formula:
((13-1)/2)/2 *[ 2*(1+4) + ((13-1)/2 - 1)*2*4]
= 12/4 * [2*5 + (12/2 - 1)*8]
= 3 * [10 + 5 * 8]
= 3 * 50
Answer= 150.
Step-by-step explanation:
The sum, S of an A.P. series is given by formula:
S = number of terms/2*[2* first term + (number of terms - 1) * common difference]
Therefore,
The sum of the odd numbered terms, SO:
SO = ((n+1)/2)/2*[2*1 + ((n+1)/2 - 1)* 2*d] = 175
=> (n+1)*[2 + (n+1)*d - 2*d] = 4* 175 = 700 — Eqn. I
The sum of the even numbered terms, SE:
SE = ((n-1)/2)/2 *[ 2*(1+d) + ((n-1)/2 - 1)*2*d] = 150
=> (n-1)*[2 + 2*d + (n-1)*d - 2*d = 150 * 4 = 600 — Eqn. II
If you solve Eqn. I & Eqn. II for n and d simultaneously, you will get n = 13, and d = 4.
Therefore, the original series has 13 terms starting at 1, with a common difference of 4:
1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 + 37 + 41 + 45 + 49.
The sum:
1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 + 37 + 41 + 45 + 49 = 325.
By formula:
S = 13/2*[2*1 + (13–1)* 4]
= 13/2*(2 + 12*4)
= 13/2*(50)
= 325.
Sum of the odd numbered terms: 1+ 9 + 17 + 25 + 33 + 41 + 49 = 175.
And by formula:
((13+1)/2)/2 *[2 + ((13+1)/2 -1) * 2 * 4]
= 14/4*[2 + (14/2 - 1)*8]
= 14/4*[2 + 6*8]
= 14/4 * (50)
= 175.
Sum of the even numbered terms: 5 + 13 + 21 + 29 + 37 + 45 = 150.
EXPLAINATION;
We need to find the number of terms, n and the common difference, d of the A.P. series of odd numbers: 1 + (1 + (2-1)d ) + (1 + (3–1)d ) + (1 + (4–1)d) + …. + (1 + (n-1)d, since the nth term, tn is given by:
tn = t1 + (n-1)d = 1 + (n-1)d, and t1 = 1.
Since the sum of the odd numbered terms is 175, and the sum of the even numbered terms is 150, then the sum of the entire series of odd numbers is 175 + 150 = 325.
Since the sum of the series is odd and all the terms are odd, then the number of terms must also be odd, because the sum of an odd number of odd numbers is odd. Example: The sum of 3 odd numbers, 1 + 3 + 5 = 9, which is an odd number.
Therefore, the number of odd numbered terms should be one more than the number of even numbered terms. If the number of terms of the original series is n, then the number of the even numbered terms is (n -1)/2, and the number of odd numbered terms is (n-1)/2 + 1 = (n + 1)/2.
Since the terms are odd, the common difference, d in the original series must be even, and the common differences of the series of the odd numbered terms and that of the even numbered terms must be 2 times d (2*d).
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