Question

If five times the fifth term of an ap=eight times its eighth term then show that it's thirteenth term is 0

Answer 1

Answer:

QED

Step-by-step explanation:

5th term = a +4d

8th term = a + 7d

13th term = a +12d

5(a+4d) = 8(a+7d)

5a + 20d = 8a + 56d

- 36d = 3a

-12d = a

a + 12d = 0

13th term = a + 12d = 0

QED

Answer 2

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: SOLUTION \: \: } \mid}}}}}$

$\underline{ \mathfrak{ \: \: Given, \: \: }} \\\ \\ \mathtt{ \rightsquigarrow 5 \: times \: \: of \: \: the \: \: 5th \: \: term = 8th \: \:times} \\ \mathtt{ \: \: \: \: \:\: \: \: \: \:\: \: \: \: \: \:\: \: \: \: \: \: \: \: \:\: \: \:\: \: \:\: \: \: \: of \: \: the \: \: 8th \: \: term} \\ \\ \\ \underline{ \mathfrak{ \: \: </p><p>To \: show : \mapsto \: \: }} \\ \\ \mathtt{ \rightsquigarrow \: 13th \: \: term \: \: is \: \: equal \: \: to \: \: zero. \: }$

$\underline{ \mathfrak{ \: \: Suppose, \: \: }} \\ \\ \mathtt{ \rightsquigarrow \: First \: \: term \: \: of \: \: the \: \: A.P. = a} \\ \mathtt{ \rightsquigarrow \: Difference = d}$

$\underline{ \bold{ \: A.T.Q., \: }} \\ \\ \: \: \: \: \: \: \mathtt{ 5 \times T_5 = 8 \times T_8 }\\ \\ \mathtt{\Longrightarrow \: 5[ a + (5-1)d]=8 [ a + (8-1)d]} \\ \\ \mathtt{\Longrightarrow 5(a + 4d)= 8 (a+7d) }\\ \\ \mathtt{ \Longrightarrow 5a + 20 \: d = 8a + 56 \: d} \\ \\ \mathtt{ \Longrightarrow 5a - 8a = 56 \: d - 20 \: d}\\ \\ \mathtt{\Longrightarrow - 3a = 36 \: d}\\ \\ \mathtt{\Longrightarrow a = \frac{36 \: d}{ - 3}} \\ \\ \: \: \: \mathtt{ \therefore \: \: \underline{ \: a = - 12 \: d \: }}</p><p></p><p>$

$\underline{\mathfrak{ \: Now \: }} \\ \\ \mathtt {13th \: \: term, \: T_{13 }= a + [(13-1)d] } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \mathtt{ = a + 12 \: d} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \mathtt{ = ( - 12 \: d) + 12 \: d } \\ \mathtt{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{[As \: \: a = - 12 \: d]}} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\mathtt{ = - 12 \: d + 12 \: d} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \mathtt{ = 0} \\ \\ \\ \huge{\mathtt{ \therefore \: \: \underline{ \: \: 13th \: \: term \: = 0 \: \: }} }\\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: </p><p> \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \mathtt{ \underline{ \: Hence \: \: proved. \: \: }}$