If the sum of seven consecutive even integers is 140, then what is the largest even integer among them? 

यदि सात क्रमिक सम पूर्णांकों का योग 140 है तो इनमे से सबसे बड़ा सम पूर्णांक क्या है ?

A..(22)

B..(24)

C..(26)

D..(28)

Answer C.

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Solution:

I Method:

Let 7 consecutive even integers be x, x+2, x+4, x+6, x+8, x+10, x+12.

As per question -

x, x+2, x+4, x+6, x+8, x+10, x+12 = 140

7x + 42 = 140

7x = 140 - 42

7x = 98

x = 98/7

x = 14

Largest even integer x+12 = 14+12 = 26

II Method:

If the sum of n (n is always odd) consecutive numbers is known, then the exact middle number = sum/n

As per question -

The exact middle number = 140/7 = 20

The exact middle number or fourth number = 20

Since we are given the sum of seven consecutive even integers and we have found the fourth number, then we will get the next numbers by adding 2 to the fourth number.

Largest even integer or seventh number = fourth number + 2 + 2 + 2 = 20 + 2 + 2 + 2 = 26

Hence, the largest even integer = 26

Hence, the correct answer is option C.

हल:

I Method:

माना 7 क्रमागत सम पूर्णाक x, x+2, x+4, x+6, x+8, x+10, x+12 है l

प्रश्नानुसार -

x, x+2, x+4, x+6, x+8, x+10, x+12 = 140

7x + 42 = 140

7x = 140 - 42

7x = 98

x = 98/7

x = 14

सबसे बड़ा सम पूर्णाक x+12 = 14+12 = 26

II Method:

यदि n (n हमेशा विषम हो) क्रमागत संख्याओं का योग ज्ञात हो तब ठीक बीच वाली संख्या = योग/n

प्रश्नानुसार -

ठीक बीच वाली संख्या = 140/7 = 20

ठीक बीच वाली संख्या या चौथी संख्या = 20

चूँकि हमें सात क्रमिक सम पूर्णाक संख्याओं का योग दिया हुआ है और हमने चौथी सख्या ज्ञात कर ली है तब हमें अगली संख्यायें चौथी संख्या में 2 जोड़ने पर प्राप्त होंगी l

सबसे बड़ा सम पूर्णाक या सातवीं संख्या = चौथी संख्या + 2 + 2 + 2 = 20 + 2 + 2 + 2 = 26

अतः सबसे बड़ा सम पूर्णाक = 26

अतः सही उत्तर विकल्प C है l

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