Question

Find three numbers in AP whose sum is 24 and the sum of whose squares is 224

Answer 1

$\large\bf\underline{Given:-}$

• Sum of terms = 24
• sum of squares of terms is 224

$\large\bf\underline {To \: find:-}$

• terms

$\huge\bf\underline{Solution:-}$

Let the three terms be (a-d ), a , (a+ d)

According to question :-

First condition:-

• sum of three terms is 24

»» (a - d) + a ( a + d ) = 24

»» a - d + a + a + d = 24

»» 3a = 24

»» a = 24 /3

»» a = 8

Second condition :-

• Sum of squares of three terms is 224

»» (a - d)² + a² + ( a+ d)² = 224

»» a² + d² - 2ad + a² + a² + d² + 2ad = 224

»» a² + a² + a² + d² + d² = 224

»» 3a² + 2d² = 224

• putting value of a = 8

»» 3 × 8² + 2d² = 224

»» 3 × 64 + 2d² = 224

»» 192 + 2d² = 224

»» 2d² = 224 - 192

»» 2d² = 32

»» d² = 32/2

»» d² = 16

»» d = √16

»» d = 4

So,

three numbers are :-

a - d = 8 - 4 = 4

a = 8

a + d = 8 + 4 = 12.

✝️Three numbers are 4 , 8 , 12

Answer 2

$\rule{200}{2}$

$\Huge\bigstar\:\tt\underline\red{GIVEN}$

$\:\:\:\:\:\bullet\:\:\:\:\sf\blue{ Sum \:of \:terms\:=\:24}$

$\:\:\:\:\:\bullet\:\:\:\:\sf\blue{ Sum\: of\: squares \: of\: terms\:=\:224}$

$\rule{200}{2}$

$\Huge\bigstar\:\tt\underline\red{TO\:FIND}$

$\:\:\:\:\:\:\:\:\:\:\:\:\bullet\:\:\:\:\sf\blue{Terms}$

$\rule{200}{2}$

$\Huge\bigstar\:\tt\underline\red{SOLUTION}$

$\sf\orange{Let \: (a - d),\: a, \:(a + d) \:be\: the\: three \:terms \:of \:AP}$

$\sf\gray{sum \:of \:these \:three\: terms}:$

$\Large\leadsto \:\:\:\: \sf \purple {a - d + a + a + d = 24}$

$\Large\leadsto \:\:\:\: \sf \green {3a = 24}$

$\Large\leadsto \:\:\:\: \sf \purple {a={\huge\frac{24}{3}} }$

$\Large\leadsto \:\:\:\: \sf \green {a = 8}$

$\large\star\:\sf\underline\pink{Sum\: of \:square \:of\: these\: terms} :$

$\Large\leadsto \:\:\:\: \sf \purple {(a-d)^{2}+a^{2}+(a+d)^{2}= 224}$

$\Large\leadsto \:\:\:\: \sf \green {a^{2}+d^{2}-2ad+a^{2}+a^{2}+d^{2}+2ad=224}$

$\Large\leadsto \:\:\:\: \sf \purple {3a^{2}+2d^{2}=224}$

$\large\star\:\sf\underline\pink{Plugging\: a \:=\: 8}$

$\Large\leadsto \:\:\:\: \sf \purple {3 × 8^{2} + 2d^{2} = 224}$

$\Large\leadsto \:\:\:\: \sf \green {3 × 64 + 2d^{2} = 224}$

$\Large\leadsto \:\:\:\: \sf \purple {192 + 2d^{2} = 224}$

$\Large\leadsto \:\:\:\: \sf \green {2d^{2} = 224 - 192}$

$\Large\leadsto \:\:\:\: \sf \purple {2d^{2} = 32}$

$\Large\leadsto \:\:\:\: \sf \green {d^{2} = {\huge\frac{32}{2}} }$

$\Large\leadsto \:\:\:\: \sf \purple {d^{2} = 16}$

$\Large\leadsto \:\:\:\: \sf \green {d = \sqrt{16}}$

$\Large\leadsto \:\:\:\: \sf \purple {d = 4}$

$\Large\sf\underline\orange{So, \: 3 \:numbers\: are} :-$

$\Large\leadsto \:\:\:\: \sf \purple {a - d = 8 - 4 = 4}$

$\Large\leadsto \:\:\:\: \sf \green {a = 8}$

$\Large\leadsto \:\:\:\: \sf \purple {a + d }$

$\Large\leadsto \:\:\:\: \sf \green { 8 + 4}$

$\Large\leadsto \:\:\:\: \sf \purple {12}$

$\large\dagger\:\:\:\bf\underline\red{Three\:\:Numbers\:\:Are}$

• $\Large\mathfrak\pink{4}$
• $\Large\mathfrak\pink{8}$
• $\Large\mathfrak\pink{12}$

$\rule{200}{2}$