and =N +(2n 1) n-. 1 +3+5+ 13. The identity (1+2+3++n) n 1 = 13 +23 +33 + .+ n3, as the first century. Provide a was known as early derivation of it. 14. Prove the following formula for the sum of triangular numbers, given by the Hindu mathematician Aryabhata (circa 500): n(n +1)(n+2) t1t2 +t3 + +tn Hint: Group the terms on the left-hand side in pairs, replacing t-1 + tk by k; consider the two cases where n is odd and n is even.] 15. Archimedes (287-212 B.C.) also derived the formula 12+22+32 + n(n + 1)(2n + 1) + n2 11 for the sum of squares. Fill in any missing details in the 4 Prove ttt+t3+ -- -tttn hCntl)(n+2) Hint: Group the terms in LHS in pais, replacing tx ttx wlK (csnsider n is odd or even) Suppose n is even: (tt) (t3+t) (tstto+tntn) 122232+ tn 2. 2. 4 2 ча 2 nin+1)12n+) Lo 2t4+ 2 (ak)2 + n ak k2 4C12+22+3 -- - + K2 4k(k+) (2k+1) Z igure us nint (nt2) la and smailar todd t n is odd t,ttztts+tytta tn-tth It3 +52+ t35 2422132142+5t+(k2+2K1)2 -(22+42t2(2k) ak+)ak+ 1} +1)(al2ki) 4K) akt S+D90 00rtsvnsts Sbwey nakt 2. +n2 2+ dd im forms missirg ble we ceunfaetsr anything out nintlnt) 1 11 I ( l
and =N +(2n 1) n-. 1 +3+5+ 13. The identity (1+2+3++n) n 1 = 13 +23 +33 + .+ n3, as the first century. Provide a was known as early derivation of it. 14. Prove the following formula for the sum of triangular numbers, given by the Hindu mathematician Aryabhata (circa 500): n(n +1)(n+2) t1t2 +t3 + +tn Hint: Group the terms on the left-hand side in pairs, replacing t-1 + tk by k; consider the two cases where n is odd and n is even.] 15. Archimedes (287-212 B.C.) also derived the formula 12+22+32 + n(n + 1)(2n + 1) + n2 11 for the sum of squares. Fill in any missing details in the 4 Prove ttt+t3+ -- -tttn hCntl)(n+2) Hint: Group the terms in LHS in pais, replacing tx ttx wlK (csnsider n is odd or even) Suppose n is even: (tt) (t3+t) (tstto+tntn) 122232+ tn 2. 2. 4 2 ча 2 nin+1)12n+) Lo 2t4+ 2 (ak)2 + n ak k2 4C12+22+3 -- - + K2 4k(k+) (2k+1) Z igure us nint (nt2) la and smailar todd t n is odd t,ttztts+tytta tn-tth It3 +52+ t35 2422132142+5t+(k2+2K1)2 -(22+42t2(2k) ak+)ak+ 1} +1)(al2ki) 4K) akt S+D90 00rtsvnsts Sbwey nakt 2. +n2 2+ dd im forms missirg ble we ceunfaetsr anything out nintlnt) 1 11 I ( l
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Topic Video
Question
Number 14 please.
Got partly through the problem
![and
=N
+(2n 1) n-.
1 +3+5+
13. The identity
(1+2+3++n)
n 1
= 13 +23 +33 + .+ n3,
as the first century. Provide a
was known as early
derivation of it.
14. Prove the following formula for the sum of triangular
numbers, given by the Hindu mathematician
Aryabhata (circa 500):
n(n +1)(n+2)
t1t2 +t3 + +tn
Hint: Group the terms on the left-hand side in pairs,
replacing t-1 + tk by k; consider the two cases where
n is odd and n is even.]
15. Archimedes (287-212 B.C.) also derived the formula
12+22+32 +
n(n + 1)(2n + 1)
+ n2
11
for the sum of squares. Fill in any missing details in the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc27b03fb-27ff-4129-a29b-052f3c3c0c2f%2Ff2f05ee1-7c86-40f7-a846-d3ba054e7acd%2Ftugeqzb.jpeg&w=3840&q=75)
Transcribed Image Text:and
=N
+(2n 1) n-.
1 +3+5+
13. The identity
(1+2+3++n)
n 1
= 13 +23 +33 + .+ n3,
as the first century. Provide a
was known as early
derivation of it.
14. Prove the following formula for the sum of triangular
numbers, given by the Hindu mathematician
Aryabhata (circa 500):
n(n +1)(n+2)
t1t2 +t3 + +tn
Hint: Group the terms on the left-hand side in pairs,
replacing t-1 + tk by k; consider the two cases where
n is odd and n is even.]
15. Archimedes (287-212 B.C.) also derived the formula
12+22+32 +
n(n + 1)(2n + 1)
+ n2
11
for the sum of squares. Fill in any missing details in the
Transcribed Image Text:4
Prove
ttt+t3+ -- -tttn
hCntl)(n+2)
Hint: Group the terms in LHS in pais, replacing
tx ttx
wlK (csnsider n is odd or even)
Suppose n is even: (tt) (t3+t) (tstto+tntn)
122232+
tn
2.
2.
4 2
ча
2
nin+1)12n+)
Lo
2t4+
2
(ak)2
+
n ak
k2
4C12+22+3 -- - + K2
4k(k+) (2k+1)
Z
igure us
nint (nt2)
la
and smailar todd
t
n is odd
t,ttztts+tytta tn-tth
It3 +52+
t35
2422132142+5t+(k2+2K1)2
-(22+42t2(2k)
ak+)ak+ 1} +1)(al2ki) 4K) akt
S+D90 00rtsvnsts
Sbwey
nakt
2.
+n2
2+
dd im forms
missirg ble
we ceunfaetsr
anything out
nintlnt)
1
11
I
(
l
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
