Question

If pth term of an A. P. is 1/q and qth term is 1/p, prove that the sum of the first pq terms is 1/2(pq+1)​

Answer 1

Step-by-step explanation:

Follow the attachment for explanation.

Answer 2

GIVEN:-

Ap = 1/q

Aq = 1/p

TO PROVE:-

Spq = 1/2(pq + 1 )

PROOF:-

Aq = a + (n-1)×d

1/p = a + (q-1)×d. {as Aq = 1/p} ----(1)

Ap = a + (n - 1)×d

1/q = a + (p-1)×d. {asAp = 1/q}. ------(2)

SUBTRACT EQUATION (1) FROM (2)

1/q = a + (p-1)×d

1/p = a + (q-1)×d

- - -

____________

1/q - 1/p = 0 + d{p-1-q+1}

p-q/pq = d{p-q}

1/pq = d

SUBSTITUTE VALUE OF d IN EQUATION (2)

1/q = a + (p-1)× 1/pq

1/q = a + p/pq -1/pq

1/q = a + 1/q - 1/pq

1/q - 1/q + 1/pq = a

1/pq = a

Sn = n/2 {2a + (n-1)×d}

Spq = pq/2{2a + (pq-1)×d}

substitute Value of a and d in above equation

Spq ko = pq/2 {2 × 1/pq + (pq-1)×1/pq}

Spq = pq/2 {2/pq + pq/pq - 1/pq }

Spq = pq/2{2/pq + 1 - 1/pq }

Spq = pq/2 {2/pq - 1/pq + 1}

Spq = pq/2{1/pq + 1}

Spq = pq/2{1 + pq/pq}

Spq = pq/2 {1 + pq}/pq

Spq = 1/2{1 + pq}

HENCE PROVED

_____________________________________