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Question
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Solution 1
The given sequence is 2, 4, 6, 8 ... 344, 346, 348
Now, surely the above-mentioned range is in an A.P. or Arithmetic Progression. (Because the difference between two consecutive numbers in the given series is always constant.)
The above sequence is an A.P. with
a = t1 = 2,
d = t2 - t1
= 4 - 2
= 2
tn = 348
Since tn = a + (n - 1) d
∴ 348 = 2 + (n - 1) 2
∴ 348 = 2 - 2n + 2
∴ 348 = 2 + 2n - 2
∴ 348 = 2n
∴ n = `348/2`
∴ n = 174
Thus, the number of terms (n) = 174.
Now, Sn = `n/2` [2a + (n - 1)d]
∴ S174 = `174/2` [2(2) + (174 - 1)2]
= 87 [4 + (173)2]
= 87 [4 + 346]
= 87 × 350
= 30450
Hence, the sum of all even numbers between 1 and 350 is 30450.
Solution 2
All even numbers between 1 and 350,
2, 4, 6, 8 ... 344, 346, 348
Given sequence is an A.P.
a = 2, d = 2, tn = 348
tn = a + (n - 1)d
348 = 2 + (n - 1)2
348 - 2 = (n - 1)2
`346/2` = n - 1
173 + 1 = n
n = 174
t1 = 2, tn = 348
Sn = `n/2[t_1+t_n]`
S174 = `174/2[2+348]`
S174 = 87 × 350
S174 = 30450
Hence, the sum of all even numbers between 1 and 350 is 30450.
RELATED QUESTIONS
Determine the A.P. whose 3rd term is 16 and the 7th term exceeds the 5th term by 12.
In an AP given a = 8, an = 62, Sn = 210, find n and d.
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If the pth term of an AP is q and its qth term is p then show that its (p + q)th term is zero.
The first and last terms of an AP are a and l respectively. Show that the sum of the nth term from the beginning and the nth term form the end is (a + l).
Write the next term of the AP `sqrt(8), sqrt(18), sqrt(32),`....
If (2p – 1), 7, 3p are in AP, find the value of p.
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The next term of the A.P. \[\sqrt{7}, \sqrt{28}, \sqrt{63}\] is ______.
First term and the common differences of an A.P. are 6 and 3 respectively; find S27.
Solution: First term = a = 6, common difference = d = 3, S27 = ?
Sn = `"n"/2 [square + ("n" - 1)"d"]` - Formula
Sn = `27/2 [12 + (27 - 1)square]`
= `27/2 xx square`
= 27 × 45
S27 = `square`
Choose the correct alternative answer for the following question .
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The 9th term of an A.P. is 449 and 449th term is 9. The term which is equal to zero is
Q.5
In an A.P. sum of three consecutive terms is 27 and their products is 504. Find the terms. (Assume that three consecutive terms in an A.P. are a – d, a, a + d.)
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Find the next 4 terms of the sequence `1/6, 1/4, 1/3`. Also find Sn.
The sum of the first 15 multiples of 8 is ______.
Show that the sum of an AP whose first term is a, the second term b and the last term c, is equal to `((a + c)(b + c - 2a))/(2(b - a))`
Jaspal Singh repays his total loan of Rs. 118000 by paying every month starting with the first instalment of Rs. 1000. If he increases the instalment by Rs. 100 every month, what amount will be paid by him in the 30th instalment? What amount of loan does he still have to pay after the 30th instalment?
