Question
Using mathematical induction, prove that the sum of the squares of first 'n' natural numbers is .
Answer :
Word Count : 204
________ __________ _____ _______ ________ ________ __________ _______ __________ __________ _______ _______.
______ _______ ____ ____ _______ ___ __________ ___ ___ ________.
______ _____ _______ __________ ________ _____ ________ ______.
___ ___ ___ __________ ____ ______ ________ ______ ____.
__________ ____ ____ _________ __________ ________.
_________ ______ ________ ____ _______.
_________ _______ __________ __________ _____ ________ __________.
__________ _____ __________ ___ _______ _________ _____ ____ _________ ________ _______.
_________ ____ ______ __________ ____ _______ ________ ___ _______.
_____ ____ ______ ________ __________ _____ ______ _________ _______ _________ __________ _____.
______ ________ ____ ________ __________ ______ ________ _____ ___.
___ __________ ___ ___ _____ ___ ________ ___ ____ ___.
_________ _______ _________ ________ __________.
_____ _________ ________ ________ _______ ____ _____.
Skip in 10
For (n = 1), the sum of the squares of the first natural number is (1^2 = 1). The given formula gives (\frac{1(1+1)(2\cdot1+1)}{6} = \frac{1\cdot2\cdot3}{6} = 1). Hence the result is true for (n = 1). Assume that the formula is true for some natural number (n = k), that is, [ 1^2 + 2^2 + 3^2 + \cdots + k^2 = \frac{k(k+1)(2k+1)}{6}. ] Now consider the sum of squares up to (k+1):
______ __________ ________ __________ ________ _______ __________ __________ _______ ___ __________.________ __________ _____ _______ ________ ________ __________ _______ __________ __________ _______ _______.
______ _______ ____ ____ _______ ___ __________ ___ ___ ________.
______ _____ _______ __________ ________ _____ ________ ______.
___ ___ ___ __________ ____ ______ ________ ______ ____.
__________ ____ ____ _________ __________ ________.
_________ ______ ________ ____ _______.
_________ _______ __________ __________ _____ ________ __________.
__________ _____ __________ ___ _______ _________ _____ ____ _________ ________ _______.
_________ ____ ______ __________ ____ _______ ________ ___ _______.
_____ ____ ______ ________ __________ _____ ______ _________ _______ _________ __________ _____.
______ ________ ____ ________ __________ ______ ________ _____ ___.
___ __________ ___ ___ _____ ___ ________ ___ ____ ___.
_________ _______ _________ ________ __________.
_____ _________ ________ ________ _______ ____ _____.
IGNOU ACADEMY
Your IGNOU journey, made easier.
- Solved Assignments
- Study Notes
- Guess Papers
- Question Papers
Continue Reading — It's Free
Sign in with Google to unlock more of this answer.
✓ Free access
✓ Unlock up to 60% of the answer
✓ No payment required
✓ Unlock up to 60% of the answer
✓ No payment required
Quick Google sign-in. No separate registration form.
Related Questions
बोलजनों-वेयरस्ट्रास प्रमेय लिखिए। इसका प्रयोग करके जाँच कीजिए कि समुच्चय का कोई सीमा-बिंदु है या नहीं।
Suppose that f and g are in R[a, b]. Then show that the function f + g is in R[a, b], and
Discuss the convergence of the p-series by using the integral test.
मान लीजिए कि f तथा g, R[a, b] में हैं। तब दिखाइए कि फलन f + g भी R[a, b] में हैं, तथा .
मान लीजिए कि f एक अवकलनीय फलन है, जिसका अवकलज अंतराल [a, b] पर शून्य नहीं होता। तब दिखाइए कि f या तो सर्वथा वर्धमान है अथवा सर्वथा ह्रासमान है।
Prove that every convergent sequence is Cauchy.
फलन f की पर असांतत्य के प्रकार की जाँच कीजिए, जहाँ
है।