Question

If $\frac{5 + 9 + 11 + ...to \: n \: terms}{7 + 9 + 11 + ...to \: (n + 1) \: terms} = \frac{17}{16}$, then n = ?
(a)8
(b)7
(c)10
(d)11
Answer with proper explanation.
Question from Class-10 Chapter:– Arithmetic Progressions.​

Answer 1

Correct Question

If $\frac{5 + 9 + 13 + ...to \: n \: terms}{7 + 9 + 11 + ...to \: (n + 1) \: terms} = \frac{17}{16}$, then n = ?

(a)8

(b)7

(c)10

(d)11

Solution -

For n terms:

Sn = n/2 (2a + (n -1)d)

5+9+13+---- (n term)

a = 5, d = 9-5 = 4

Sn = n/2 (2*5 + (n-1)4)

Sn = n/2 (10 + 4n - 4)

Sn = n/2 (4n + 6)

Sn = n(2n + 3) ............(1st equation)

For nth term

S(n+1) = (n+1)/2 (2a + (n+1-1)d)

7+9+11-----(n+1 term)

a = 7, d = 9-7 = 2

S(n+1) = (n+1)/2 (2*7 + (n)2)

S(n+1) = (n+1)/2 (14 + 2n)

S(n+1) = (n+1)(7 + n) .............(2nd equation)

Fraction of both 1st & 2nd equation is 17/16 (given in question)

n(2n+3)/(n+1)(7+n) = 17/16

Cross-multiply them

16n(2n+3) = 17(n+1)(7+n)

32n² + 48n = (17n+17)(7+n)

32n² + 48n = 119n + 17n² + 119 + 17n

32n² + 48n = 136n + 17n² + 119

15n² - 88n - 119 = 0

The above equation is in the form ax² + bx + c = 0

Now, solve it by splitting the middle term.

15n² - 105n + 17n - 119 = 0

15n(n - 7) +17(n - 7) = 0

(n - 7)(15n + 17) = 0

n = +7, -17/15

Negative one neglected. We left with +7.

Therefore, n = 7

Option b) 7

Answer 2

SoluTion:

Correct expression :

• (5 + 9 + 13 + .... n terms)/(7 + 9 + 11 + ....n + 1 terms) = 17/16

We've to find the value of n.

Consider 1st AP : 5 + 9 + 13 + ... n terms

Sum = 17 K

Consider 2nd AP : 7 + 9 + 11 + ...(n + 1) terms

Sum = 16 K

____________

S1 = n/2 [ 2 × 5 + {n - 1) 4]

→ 17K = n (3 + 2n) ........(1)

S2 = (n+1)/2 [ 2 × 7 + 2n ]17

→ 16K = (n + 1)(7 + n) ........(2)

Dividing (1) by (2).

→ [n(3 + 2n) = 17K]/[(n + 1)(7 + n) = 16K]

Cross multiplying it,

→ 17(n + 1)(7 + n) = 16n(3 + 2n)

→ 17(n² + 8n + 7) = 48n + 32n²

→ 17n² + 136n + 119 = 48n + 32n²

→ 17n² - 32n² + 136n - 48n + 119 = 0

→ -15n² + 88n + 119 = 0

Using splitting middle term method, we get,

n = 7 and -17/15

Rejecting the negative value, we get,

→ n = 7

Hence, option B) is correct.