Below are a compilation of four letters by Srila Prabhupada further elucidating this subject.
Regarding your third question—"Why are there apparent time-gaps in the line of disciplic succession as listed in the Bhagavad-gita? Is Arjuna an instructor Spiritual Master and not an initiator Spiritual Master and therefore not listed?'' The time gap mentioned by you is inevitable, because the disciplic succession sometimes becomes disconnected, as we find from the Bhagavad-gita. This is the influence of material energy, and to link it up again, it takes some time. That some time may appear to our calculation a big gap, but in relation with the eternal time, it is not even an instant. So this big gap or small gap of time is relative. Just like our 24 hours and Brahma's 24 hours, there is much difference. Our 24 hours is not even a fraction of his second....
- Letter to Rupanuga, March 14, 1969, Hawaii
Regarding parampara system: there is nothing to wonder for big gaps. Just like we belong to the Brahma Sampradaya, so we accept it from Krishna to Brahma, Brahma to Narada, Narada to Vyasadeva, Vyasadeva to Madhva, and between Vyasadeva and Madhva there is a big gap. But it is sometimes said that Vyasadeva is still living, and Madhva was fortunate enough to meet him directly. In a similar way, we find in the Bhagavad-gita that the Gita was taught to the sun god, some millions of years ago, but Krishna has mentioned only three names in this parampara system—namely, Vivasvan, Manu, and Ikshvaku; and so these gaps do not hamper from understanding the parampara system. We have to pick up the prominent acharya, and follow from him. There are many branches also from the parampara system, and it is not possible to record all the branches and sub-branches in the disciplic succession. We have to pick up from the authority of the acharya in whatever sampradaya we belong to. Hoping you are all three well, and please keep me informed.
- Letter to Dayananda, April 12, 1968, San Francisco
"Narada was the Spiritual Master of Vyasadeva, and Arjuna was Vyasadeva's disciple, not as initiated disciple but there was some blood relation between them. So there is connection in this way, and it is not possible to list all such relationships in the short description given in Bhagavad-gita As It Is. Another point is that disciplic succession does not mean one has to be directly a disciple of a particular person. The conclusions which we have tried to explain in our Bhagavad-gita As It Is is the same as those conclusions of Arjuna. Arjuna accepted Krishna as the Supreme Personality of Godhead, and we also accept the same truth under the disciplic succession of Chaitanya Mahaprabhu. Things equal to the same thing are equal to one another. This is an axiomatic truth. So there is no difference of opinion of understanding Krishna between ourselves and Arjuna. Another example is that a tree has many branches, and you will find one leaf here and another leaf there. But if you take this leaf and the other leaf and you press them both, you will see that the taste is the same. The taste is the conclusion, and from the taste you can understand that both leaves are from the same tree..."
- Letter to Kirtanananda, January 25, 1969, Los Angeles
Regarding the disciplic succession coming from Arjuna, disciplic succession does not always mean that one has to be initiated officially. Disciplic succession means to accept the disciplic conclusion. Arjuna was a disciple of Krishna and Brahma was also a disciple of Krishna. Thus there is no disagreement between the conclusions of Brahma and Arjuna. Vyasadeva is in the disciplic succession of Brahma. The teachings to Arjuna was recorded by Vyasadeva verbatim. So according to the axiomatic truth, things equal to one another are equal to each other. We are not exactly directly from Vyasadeva, but our Gurudeva is a representative of Vyasadeva. Because Vyasadeva and Arjuna are of equal status, being students of Krishna, therefore we are in the disciplic succession of Arjuna. Things equal to the same thing are equal to one another...
- Letter to Dinesh, October 31, 1969, Tittenhurst